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MATHEMATICAL   MONOGRAPHS. 

EDITED  BY 

MANSFIELD   MERRIMAN  AND  ROBERT  S.  WOODWARD. 


No.  12. 


THE    THEORY 


OF 


RE  L  AT  I  VITY 


BY 


ROBERT  D.  CARMICHAEL, 

PROFESSOR  OF  MATHEMATICS  IN  UNIVERSITY  OF  ILLINOIS 


SECOND     EDITION 


NEW   YORK 

JOHN    WILEY   &   SONS,    INC. 

LONDON:  CHAPMAN  &  HALL,  LIMITED 
1920 


Engineering 
Library 


COPYRIGHT,  1913.  1920, 

BY 
ROBERT   D.  CARMICHAEL. 


PRESS  or 

BRAUNWOHTH    &   CO. 

BOOK  MANUFACTURERS 

BROOKLYN,   N.  V. 


PREFACE 


THE  theory  of  relativity  has  now  reached  its  furthest  con- 
ceivable generalization  in  the  direction  of  the  covariance  of  the 
Jaws  of  nature  under  transformations  of  coordinates.  The 
older  theory  of  relativity  remains  valid  as  a  special  case  of  the 
general  theory  and  may  well  serve  as  an  introduction  to  its 
more  far-reaching  aspects.  Accordingly,  in  the  present  (secona) 
edition  of  this  monograph  I  have  retained  the  older  theory  in 
precisely  the  same  form  as  in  the  first  edition,  the  matter  cover- 
ing Chapters  I  to  VI  of  the  present  treatment,  and  have  added 
the  longer  Chapter  VII  to  give  a  compact  account  of  the  general- 
ized theory.  The  tendency  now  is  to  call  the  latter  the  theory 
of  relativity  and  to  distinguish  the  older  from  it  by  giving  to 
the  older  theory  the  name  of  the  restricted  theory  of  relativity. 

In  the  opening  section  (§37)  of  the  new  chapter,  I  give  a 
brief  summary  of  results  from  the  restricted  theory.  Anyone 
who  is  acquainted  with  these,  whether  derived  as  in  this  book 
or  otherwise,  may  proceed  at  once  to  the  reading  of  Chapter 
VII.  It  is  believed  that  he  will  find  in  it  about  as  brief  an  account 
of  the  new  theory  as  can  be  given  so  as  to  be  easily  intelligible 
and  at  the  same  time  to  reach  the  general  theory  of  gravitation, 
to  make  clear  the  nature  of  the  three  famous  crucial  phenomena, 
to  associate  the  theory  with  Maxwell's  electromagnetic  equa- 
tions, and  to  place  the  whole  in  its  proper  setting  with  respect 
to  the  general  body  of  scientific  truth. 

The  new  as  well  as  the  older  matter  in  the  booklet  has  been 
written  from  the  point  of  view  of  the  usefulness  of  the  theory 
of  relativity  in  the  development  of  physical  science.  No 
applications  are  given  other  than  those  which  are  directly  and 
immediately  associated  either  with  the  fundamental  ideas  or 

3 


425854 


4*  *  PREFACE. 

with  certain  crucial  phenomena  for  testing  the  validity  of  the 
theory.  In  this  way  only  may  the  central  elements  of  novelty 
most  readily  be  brought  to  light. 

No  attempt  has  been  made  to  give  a  complete  account  of 
the  theory.  The  purpose  of  the  monograph  is  best  served  by 
presenting  only  those  fundamental  developments  which  are 
needed  for  and  contribute  directly  to  making  clear  the  main 
characteristics  of  the  theory.  The  more  detailed  statements 
are  to  be  found  elsewhere,  especially  in  the  memoirs  which  have 
now  reached  a  considerable  number. 

Every  exposition  of  the  general  theory  of  relativity  must  be 
deeply  indebted  to  the  basic  memoir  of  Einstein,  published  in 
1916  in  Annalen  der  Physik,  volume  49.  Very  useful  to  me 
also,  as  every  reader  will  observe,  has  been  the  report  of  A.  S. 
Eddington  to  the  Physical  Society  of  London  on  "The  Relativity 
Theory  of  Gravitation,"  a  booklet  to  which  one  may  be  referred 
who  wishes  to  go  further  into  the  theory  than  the  exposition  of 
the  present  monograph  will  carry  him. 

R.  D.  CARMICHAEL. 

UNIVERSITY  or  ILLINOIS, 
April,  1920. 


CONTENTS. 


CHAPTER  I.    INTRODUCTION. 

PAGE 

§    i.    THE  FOUNDATIONS  OF  PHYSICS 7 

§    2.    ARE  THE  LAWS  OF  NATURE  RELATIVE  TO  THE  OBSERVER? 8 

§    3.    THE  STATE  OF  THE  ETHER 9 

§    4.    MOVEMENT  OF  THE  EARTH  THROUGH  THE  ETHER 10 

5    5.    THE  TEST  OF  MICHELSON  AND  MORLEY 10 

§    6.    OTHER  EXPERIMENTAL  INVESTIGATIONS 13 

§    7.    THE  THEORY  OF  RELATIVITY  is  INDEPENDENT  OF  THE  ETHER 14 

CHAPTER  II.    THE  POSTULATES   OF  RELATIVITY. 

§    8.    INTRODUCTION 15 

§    g.    SYSTEMS  OF  REFERENCE 16 

§  10.    THE  FIRST  CHARACTERISTIC  POSTULATE 17 

§  ii.    THE  SECOND  CHARACTERISTIC  POSTULATE 17 

§  12.    THE  POSTULATES  V  AND  L 22 

§  13.    CONSISTENCY  AND  INDEPENDENCE  OF  THE  POSTULATES 24 

§  14.    OTHER  POSTULATES  NEEDED 25 

CHAPTER  III.    THE  MEASUREMENT  OF  LENGTH  AND  TIME. 

§  15.    RELATIONS  BETWEEN  THE  TIME  UNITS  OF  Two  SYSTEMS 27 

§  1 6.    RELATIONS  BETWEEN  THE  UNITS  OF  LENGTH  OF  Two  SYSTEMS 32 

§  17.    DISCUSSION  OF  THE  NOTION  OF  LENGTH 34 

§  18.    DISCUSSION  OF  THE  MEASUREMENT  OF  TIME 37 

§  19.    SIMULTANEITY  OF  EVENTS  HAPPENING  AT  DIFFERENT  PLACES 40 

CHAPTER  IV.    EQUATIONS  OF  TRANSFORMATION. 

§  20.    TRANSFORMATION  OF  SPACE  AND  TIME  COORDINATES 44 

§  21.    THE  ADDITION  OF  VELOCITIES 46 

§  22.    MAXIMUM  VELOCITY  OF  A  MATERIAL  SYSTEM 47 

§  23.    TIME  AS  A  FOURTH  DIMENSION 48 

5 


6  CONTENTS. 

CHAPTER  V.    MASS  AND  ENERGY. 

PAGE 

§  24.    DEPENDENCE  OF  MASS  ON  VELOCITY 49 

§  25.    ON  THE  DIMENSIONS  OF  UNITS 54 

§  26.    MASS  AND  ENERGY 57 

§  27.    ON  MEASURING  THE  VELOCITY  OF  LIGHT 58 

§  28.    ON  THE  PRINCIPLE  OF  LEAST  ACTION 59 

§  29.    A  MAXIMUM  VELOCITY  FOR  MATERIAL  BODIES 60 

§  30.    ON  THE  NATURE  OF  MASS 60 

§  31.    THE  MASS  OF  LIGHT 62 

CHAPTER  VI.    EXPERIMENTAL  VERIFICATION  OF  THE 

THEORY. 

§  32.    Two  METHODS  OF  VERIFICATION 63 

§  33.    LOGICAL  EQUIVALENTS  OF  THE  POSTULATES 65 

§  34.    ESSENTIAL  EQUIVALENTS  OF  THE  POSTULATES 65 

§  35.    THE  BUCHERER  EXPERIMENT 68 

§  36.    ANOTHER  MEANS  FOR  THE  EXPERIMENTAL  VERIFICATION  OF  THE 

THEORY  OF  RELATIVITY 70 

CHAPTER  VII.     THE  GENERALIZED  THEORY  OF 
RELATIVITY. 

§37.    SUMMARY  OF  RESULTS  FROM  PREVIOUS  CHAPTERS 73 

§38.    TRANSFORMATIONS  IN  SPACE  OF  FOUR  DIMENSIONS 75 

§  39.   THE  PRINCIPLE  OF  EQUIVALENCE 79 

§  40.    GENERAL  TRANSFORMATION  OF  AXES 81 

§  41.    THE  THEORY  OF  TENSORS 84 

§  42.    COVARIANT  DIFFERENTIATION 90 

§  43.   THE  RIEMANN-CHRISTOFFEL  TENSOR 92 

§  44.   EINSTEIN'S  LAW  OF  GRAVITATION 94 

§  45.   THE  MOTION  OF  A  PARTICLE 96 

§  46.  THREE  CRUCIAL  PHENOMENA 101 

§  47.  THE  ELECTROMAGNETIC   EQUATIONS 106 

§  48.  SOME  GENERAL  CONSIDERATIONS  RELATING  TO  THE  THEORY.  108 

INDEX..                                             .  in 


THE  THEORY  OF  RELATIVITY. 


CHAPTER  I. 
INTRODUCTION. 

§1.  THE  FOUNDATIONS  OF  PHYSICS. 

THOSE  who  look  on  physics  from  the  outside  not  infrequently 
have  the  feeling  that  it  has  forgotten  some  of  its  philosophical 
foundations.  Even  among  its  own  workers  this  condition  of 
the  science  has  not  entirely  escaped  notice. 

The  physicist,  who,  above  all  other  men,  has  to  deal  with 
space  and  time,  has  fallen  into  certain  conventions  concerning 
them  of  which  he  is  often  not  aware.  It  may  be  true  that  these 
conventions  are  just  the  ones  which  he  should  make.  It  is 
certain,  however,  that  they  should  be  made  only  by  one  who 
is  fully  conscious  of  their  nature  as  conventions  and  does  not 
look  upon  them  as  fixed  realities  beyond  the  power  of  the 
investigator  to  modify. 

Likewise,  a  question  arises  as  to  what  element  of  conven- 
tion is  involved  in  our  usual  conceptions  of  mass,  energy,  etc.; 
that  the  question  is  not  easily  answered  becomes  apparent  on 
reflection. 

These  and  many  other  considerations  suggest  the  desira- 
bility of  a  fresh  analysis  of  the  foundations  of  physical  science. 
Now  it  is  a  ground  of  gratulation  for  all  those  interested  in 
this  matter  that  there  has  arisen  within  modern  physics  itself 
a  new  movement — that  associated  with  the  Theory  of  Rela- 
tivity— which  is  capable  of  contributing  most  effectively  to  the 

7 


S  THE   THEORY  OF   RELATIVITY. 

construction  of  a  more  satisfactory  foundation  for  its  super- 
structure of  theory. 

It  is  at  once  admitted  that  the  theory  of  relativity  is  not 
yet  established  on  an  experimental  basis  which  is  satisfactory 
to  all  persons;  in  fact,  some  of  those  who  dispute  its  claim  to 
acceptance  are  among  the  most  eminent  men  of  science  of  the 
present  time.  On  the  other  hand  there  is  an  effective  body 
of  workers  who  are  pushing  forward  investigations  the  inspira- 
tion for  which  is  afforded  by  the  theory  of  relativity. 

This  state  of  affairs  will  probably  give  rise  to  a  consider- 
able controversial  literature.  If  the  outcome  of  this  contro- 
versy is  the  acceptance  in  the  main  of  the  theory  of  relativity, 
then  this  theory  will  afford  just  the  means  needed  to  arouse 
in  investigators  in  the  field  of  physics  a  lively  sense  of  the  phil- 
osophical foundations  of  their  science.  If  the  conclusions  of 
relativity  are  refuted  this  will  probably  be  done  by  a  careful 
study  of  the  foundations  of  physical  science  and  a  penetrating 
analysis  of  the  grounds  of  our  confidence  in  the  conclusions 
which  it  reaches.  This  of  itself  will  be  sufficient  to  correct 
the  present  tendency  to  forget  the  philosophical  basis  of  the 
science. 

It  follows  that  in  any  event  the  theory  of  relativity  will 
force  a  fresh  study  of  the  foundations  of  physical  theory.  If 
it  accomplishes  no  more  than  this  it  will  have  done  well. 

§  2.  ARE  THE  LAWS  OF  NATURE  RELATIVE  TO  THE  OBSERVER? 

The  fundamental  question  asked  in  the  theory  of  relativity 
is  this:  In  what  respect  are  our  enunciated  laws  of  nature  rela- 
tive to  us  who  investigate  them  and  to  the  earth  which  serves 
us  as  a  system  of  reference?  How  would  they  be  modified, 
for  instance,  by  a  change  in  the  velocity  of  the  earth? 

To  put  the  matter  more  precisely,  let  us  suppose  that  we 
have  two  relatively  moving  platforms  with  an  observer  on  each 
of  them.  Suppose  further  that  each  observer  considers  a 
system  of  reference,  say  cartesian  axes,  fixed  to  his  platform, 
and  expresses  the  laws  of  nature,  as  he  determines  them,  by  means 


INTRODUCTION.  9 

of  mathematical  equations  involving  the  cartesian  coordinates 
as  variables.  To  what  extent  will  the  laws  in  the  two  cases  be 
identical?  What  transformations  of  the  time  and  space  variables 
must  be  carried  out  in  order  to  go  from  the  equations  in  one 
system  to  those  in  the  other;  that  is,  what  relations  must  exist 
between  the  variables  on  the  two  platforms  in  order  that  the 
results  of  observation  in  the  two  cases  shall  be  consistent? 
Any  theory  which  states  these  relations  is  a  theory  of  rela- 
tivity. 

It  is  obvious  that  the  questions  above  must  be  fundamental 
in  any  system  of  mechanics.  In  fact,  a  detailed  analysis  of  the 
matter  would  show  that  such  a  system  is  characterized  pri- 
marily by  the  answers  which  it  gives  to  these  questions.  This 
is  the  feature  which  distinguishes  between  the  Newtonian  and 
the  various  systems  of  non-Newtonian  Mechanics.  The  theory 
of  relativity,  in  the  sense  of  this  book,  belongs  to  one  of  the  lat- 
ter. It  is  developed  from  a  small  number  of  fundamental 
postulates,  or  laws,  which  have  been  enunciated  as  the  probable 
teaching  of  experiment.  Some  account  of  these  experimental 
investigations  will  now  be  given. 

§3.  THE  STATE  OF  THE  ETHER. 

Those  who  postulate  the  existence  of  an  ether  as  a  means 
of  explaining  the  facts  about  light,  electricity  and  magnetism 
have  usually  been  in  general  agreement  as  to  the  conclusion 
that  the  parts  of  this  ether  have  no  relative  motion  among 
themselves,  that  is,  that  the  ether  may  be  considered  station- 
ary. Experimental  facts,  which  have  to  be  accounted  for, 
cannot  be  explained  satisfactorily  on  the  hypothesis  of  a  mobile 
ether. 

The  aberration  of  light  is  one  of  the  most  conspicuous  of 
those  phenomena  which  seem  to  require  for  their  explanation 
the  hypothesis  of  a  stationary  ether. 

The  experiment  of  Fizeau,  in  which  a  comparison  was  made 
between  the  velocities  of  light  when  going  with,  and  against, 
a  stream  of  water,  was  interpreted  by  Fresnel  as  indicating 


10  THE   THEORY   OF   RELATIVITY. 

a  certain  entrainment  of  the  ether;  but  a  later  examination 
of  the  matter  by  Lorentz  *  has  led  to  the  conclusion  that 
Fizeau's  experiment  requires  a  stationary  ether  for  its  explana- 
tion. 

A  result  which  leads  to  a  similar  conclusion  has  been 
obtained  in  electrodynamics  by  H.  A.  Wilson  f  in  measuring 
the  electric  force  produced  by  moving  an  insulator  in  a  magnetic 
field. 

§4.  MOVEMENT  OF  THE  EARTH  THROUGH  THE  ETHER. 

The  theory  of  a  stationary  ether  leads  us  to  expect  certain 
modifications  in  the  phenomena  of  light  and  electricity  when 
there  is  no  relative  motion  of  material  bodies,  but  when  both 
the  observer  and  all  his  apparatus  are  carried  along  through 
the  ether  with  a  velocity  v.  The  effects  to  be  expected  are  of 
the  order  v2/c2,  where  c  is  the  velocity  of  light.  Although 
these  effects  are  very  small  even  when  v  is  the  velocity  of  the 
earth  in  its  orbit,  the  possible  accuracy  of  certain  optical  and 
electrical  experiments  is  such  that  these  effects  could  certainly 
be  found  if  they  existed  without  some  compensating  effect 
to  mask  them.  Thus  it  should  be  possible  for  an  observer, 
by  making  optical  and  electrical  measurements  on  the  earth 
alone,  to  detect  the  motion  of  the  earth  relative  to  the  ether. 

§5.  THE  TEST  OF  MICHELSON  AND  MORLEY. 

Thus  it  was  predicted  that  the  time  which  would  be  required 
for  a  beam  of  light  to  pass  a  given  distance  and  return  would 
be  different  in  the  two  cases  when  the  path  of  light  was  parallel 
to  the  direction  of  motion  and  when  it  was  perpendicular  to 
this  direction.  Michelson  and  Morley  {  devised  an  experiment 
the- object  of  which  was  to  put  this  prediction  to  a  crucial  test. 

The  experiment  was  a  bold  one,  seeing  that  the  difference 
to  be  measured  was  so  small;  but  it  was  carried  out  in  such  a 

*  See    Lorentz,    Versuch    einer    Theorie    der    Elektrischen    und    Optischen 
Erscheinungen,  in  Bewegten  Korpern,  §  68. 
f  Proc.  Roy.  Soc.  73  (1904):  490. 
J  American  Journal  of  Science  (3),  34  (1887):  333-345. 


INTRODUCTION. 


11 


brilliant  way  as  to  permit  no  serious  doubt  of  the  accuracy  of 
the  results.  The  difference  of  time  predicted  by  theory  was 
found  by  experiment  not  to  exist;  there  was  not  the  slightest 
difference  of  time  in  the  passage  of  light  along  two  paths  of 
equal  length,  one  in  a  direction  parallel  to  the  earth's  motion 
and  the  other  in  a  direction  perpendicular  to  it. 

Owing  to  the  great  importance  which  this  famous  experiment 
has  in  the  theory  of  relativity  some  further  account  of  it  will 
be  given  here.  The  essential  parts  of  the  apparatus  used  are 
shown  in  Fig.  i,  and  the  experiment  was  carried  out  in  the  fol- 
lowing manner: 

Let  a  ray  of  light  from  a  point  source  S  fall  on  a  semi-reflect- 
ing mirror  A,  which  is  set  at  such  an  angle  that  it  will  reflect 


B  B' 


T 
FIG.  i, 


half  the  ray  to  the  mirror  B  and  allow  the  other  half  to  pass  on 
to  a  third  mirror  C.  The  lines  AB  and  AC  cross  at  right 
angles  and  the  distance  AB  is  made  equal  to  the  distance  AC. 
Half  of  the  reflected  ray  from  B  will  pass  through  A  and  on  to 
the  telescope  T.  Also,  half  of  the  reflected  ray  from  C  will  be 
reflected  at  A  to  T.  Now  the  paths  A  BAT  and  AC  AT  are 
by  measurement  equal,  so  that  the  ray  along  ABA  and  the 
one  along  AC  A  should  reach  T  simultaneously,  provided  that 
the  apparatus  is  at  rest  in  the  ether. 

Now  suppose  that  the  ether  is  stationary  and  that  the 
earth  is  moving  through  it  with  little  or  no  disturbance.  Then 
the  whole  system  of  apparatus,  which  is  fixed  to  the  earth, 
will  be  moving  with  respect  to  the  ether  with  the  effect  indicated 
in  Fig.  2. 


12  THE   THEORY  OF  RELATIVITY. 

While  the  light  is  going  from  mirror  A  to  mirrors  B  and  C 
and  back  again  to  A,  the  whole  apparatus  is  carried  forward 
in  the  direction  of  the  incident  light  to  the  position  A'B'C'. 
The  ray  reflected  from  B,  which  interferes  with  a  given  ray 
from  C  along  the  line  A'T,  must  be  considered  as  traveling 
along  the  line  AB'A',  the  angle  BAB'  being  the  angle  of  aberra- 
tion. 

Suppose  that  the  ether  remains  at  rest.  Denote  by  c  the 
velocity  of  light,  and  by  v  the  velocity  of  the  apparatus.  Let 
/  be  the  time  required  for  the  light  to  pass  from  mirror  A  to 
mirror  C,  and  let  t'  be  the  time  required  in  returning  from 
C  to  A'  .  At  the  time  when  the  reflection  takes  place  at  the 
mirror  C,  this  mirror  is  approximately  half  way  between  C  and 
C'  of  the  figure. 

Let  D  represent  the  distance  AB  or  AC  .    Then 


whence 

,_ 

- 


The  whole  time  required  for  the  passage  of  the  light  in  both 
directions  is 


and  the  distance  traveled  in  this  time  is 


the  terms  of  fourth  order  and  higher  being  neglected  in  the 
last  member. 

The  length  of  the  path  A&A'  is  evidently 

v2 


to  the  same  degree  of  accuracy  as  before.    The  difference  of 
the  two  lengths  is,  therefore,  approximately  Dv2/c2. 

If  the  whole  apparatus  is  now  turned  through  an  angle  of 
90°,  the  difference  will  be  in  the  opposite  direction,  and  hence 


INTRODUCTION.  13 

the  displacement  of  interference  fringes  along  A'T  should  be 

2Dl?  A2. 

This  is  a  very  small  difference  even  when  v  is  the  velocity 
of  the  earth  in  its  orbit;  but  it  is  altogether  sufficient  to  be 
detected  and  measured  if  it  were  present  with  no  other  effect 
to  mask  it.  The  result  of  the  experiment  was  that  practically 
no  displacement  of  interference  fringes  was  observed;  at  most 
the  displacement  was  less  than  one-fortieth  of  that  expected. 

The  conclusions  which  are  to  be  drawn  from  this  experiment 
we  shall  state  in  the  next  chapter. 

§  6.    OTHER  EXPERIMENTAL  INVESTIGATIONS. 

On  the  electrical  side  the  problem  of  detecting  the  move- 
ment of  the  earth  through  the  ether  has  been  attacked  by 
Trouton  and  Noble.*  They  hung  up  an  electrical  condenser 
by  a  torsion  wire  and  looked  for  a  torque  the  presence  of  which 
was  predicted  on  the  hypothesis  of  a  stationary  ether  through 
which  the  condenser  was  carried  by  the  motion  of  the  earth. 
Although  the  sensitiveness  of  their  electrical  arrangement  was 
ample  for  the  observation  of  the  expected  effect,  no  evidence 
of  it  was  found. 

Therefore  both  on  the  optical  and  on  the  electrical  side  the 
attempt  to  detect  the  motion  of  the  earth  through  the  ether 
fails;  no  experiment  is  known  by  which  it  can  be  put  in  evidence. 

In  addition  to  this  negative  evidence  concerning  the  pre- 
dicted effect  of  the  earth's  motion  through  the  ether  there  is 
also  the  positive  evidence  which  comes  from  the  verification 
of  contrary  predictions  based  on  other  principles.  This  will 
come  in  incidentally  for  discussion  in  our  later  chapters,  and 
consequently  will  be  dismissed  here. 

The  experiments  which  we  have  described  (and  others  related 
to  them)  are  fundamental  in  the  theory  of  relativity.  The 
postulates  of  the  next  chapter  are  based  on  them.  These 
postulates  are  in  the  nature  of  generalizations  of  the  facts 
established  by  the  experiments. 

*  Phil.  Trans.  Roy.  Soc.  (A),  202  (1904):  165. 


14  THE   THEORY  OF   RELATIVITY. 

§  7.  THE  THEORY  OF  RELATIVITY  is  INDEPENDENT  OF 
THE  ETHER. 

In  the  next  chapter  we  shall  begin  the  systematic  develop- 
ment of  the  theory  of  relativity.  It  will  be  seen  that  its 
fundamental  postulates,  or  laws,  are  based  on  the  experiments 
of  which  we  have  given  a  brief  account  and  on  others  related 
to  them.  These  experiments  have  been  carried  out  to  test 
predictions  which  have  been  made  on  the  basis  of  a  certain 
theory  of  the  ether.  But  the  results  which  have  been  obtained 
are  of  a  purely  experimental  character  and  can  be  formulated 
so  as  not  to  depend  in  any  way  on  a  theory  of  the  ether.  In 
other  words,  the  laws  stated  in  the  postulates  in  the  next 
chapter  are  in  no  way  dependent  for  their  truth  on  either  the 
existence  or  the  non-existence  of  the  ether  or  on  any  of  its 
properties. 

It  is  important  to  keep  this  in  mind  on  account  of  the 
confusion  which  has  sometimes  arisen  as  to  the  relation  between 
the  theory  of  relativity  and  the  theory  of  the  ether.  The 
postulates,  as  we  shall  see,  are  simply  generalizations  of  exper- 
imental facts;  and,  unless  an  experiment  can  be  devised  to 
show  that  these  generalizations  are  not  legitimate,  it  is  natural 
and  in  accordance  with  the  usual  procedure  in  science  to  accept 
them  as  "  laws  of  nature."  They  are  entirely  independent 
of  any  theory  of  the  ether. 


CHAPTER  II. 
THE  POSTULATES  OF  RELATIVITY. 

§  8.    INTRODUCTION. 

THERE  are  two  fundamental  postulates  concerning  the 
nature  of  space  and  time  which  underlie  all  physical  theory. 
They  assert  in  part  that  every  point  of  space  is  like  every  other 
point  and  that  every  instant  of  time  is  like  every  other  instant. 
To  make  the  statement  of  these  properties  more  exact  and 
complete  we  may  say  that  space  is  isotropic  and  homogeneous 
and  three-dimensional,  while  time  is  homogeneous  and  one- 
dimensional.  One  important  mathematical  meaning  of  this 
is  that  the  transformations  of  the  space  and  time  coordinates 
are  to  be  linear. 

All  our  theorems  will  depend  directly  or  indirectly  on  these 
two  postulates  concerning  the  nature  of  space  and  time.  Since 
it  is  certain  that  no  one  will  be  disposed  seriously  to  call  them 
in  question,  it  is  considered  unnecessary  to  give  any  further 
statement  of  them  or  to  make  explicit  reference  to  them  as  part 
of  the  basis  on  which  any  particular  theorem  depends,  it  being 
understood  once  for  all  that  they  underlie  all  our  work. 

In  the  previous  chapter  we  gave  some  account  of  the 
experiments  of  Michelson  and  Morley  and  of  Trouton  and  Noble. 
There  are  different  points  of  view  from  which  one  may  look 
at  these  experiments.  In  the  theory  of  relativity  they  are 
taken  in  the  light  of  an  attempt  to  detect  the  earth's  motion 
through  space  by  means  of  the  effect  of  this  motion  on  terrestrial 
phenomena.  So  far  as  the  experiments  go,  they  indicate  that 
such  motion  cannot  be  detected  in  this  way.  Furthermore, 
no  one  has  yet  been  able  to  devise  an  experiment  by  means  of 

15 


16  THE   THEORY   OF   RELATIVITY. 

which  the  earth's  motion  through  space  can  be  detected  by 
observations  made  on  the  earth  alone. 

The  question  arises:  Is  it  possible  to  have  any  such  exper- 
iment at  all?  In  the  theory  of  relativity  this  question  is 
answered  in  the  negative.  The  Michelson-Morley  experiment 
and  other  experiments  have  been  further  generalized  into  the 
hypothesis  that  it  is  impossible  to  detect  motion  through  space 
as  such;  that  is,  that  the  only  motion  of  which  we  can  have 
any  knowledge  is  the  motion  of  one  material  body  or  system 
of  bodies  relative  to  another.  A  sharp  formulation  of  this 
conclusion  constitutes  the  first  characteristic  postulate  of 
relativity. 

§  9.  SYSTEMS  OF  REFERENCE. 

Before  stating  the  postulate,  however,  it  will  be  necessary 
to  introduce  a  definition.  In  order  to  be  able  to  deal  with  such 
quantities  as  are  involved  in  the  measurement  of  motion,  time, 
velocity,  etc.,  it  is  necessary  to  have  some  system  of  reference 
with  respect  to  which  measurements  can  be  made.  Let  us 
consider  any  set  of  things  consisting  of  objects  and  any  kind 
of  physical  quantities  whatever  *  each  of  which  is  at  rest  with 
reference  to  each  of  the  others.  Let  us  suppose  that  among 
these  objects  are  clocks,  to  be  used  for  measuring  time,  and 
rods  or  rules,  to  be  used  for  measuring  length.  Such  a  set  of 
objects  and  quantities,  at  rest  relatively  to  each  other,  together 
with  their  units  for  measuring  time  and  length,  we  shall  call 
a  system  of  reference. f  Throughout  the  book  we  shall  denote 
such  a  system  by  S.  In  case  we  have  to  deal  at  once  with  two 
or  more  systems  of  reference  we  shall  denote  them  by  Si,  $2, 
6*3,  ...  or  by  S,  S',  .  .  .  Furthermore,  it  will  be  assumed 
that  the  units  of  any  two  systems  Si  and  $2  are  such  that 
the  same  numerical  result  will  be  obtained  in  measuring  with 

*  As,  for  instance,  charges,  magnets,  light-sources,  telescopes,  etc. 

f  If  any  number  of  these  objects  or  quantities  are  absent  we  shall  sometimes 
refer  to  what  remains  as  a  system  of  reference.  Thus  the  system  might  consist 
of  a  single  light-source  alone. 


THE   POSTULATES   OF   RELATIVITY.  17 

the  units  of  S\  a  quantity  L\  and  with  the  units  of  82  a  quantity 
Li  when  the  relation  of  L\  to  S\  is  precisely  the  same  as  that  of 
L2  to  S2. 

§  10.  THE  FIRST  CHARACTERISTIC  POSTULATE. 

With  this  definition  before  us  we  are  now  able  to  state  the 
first  characteristic  postulate  of  relativity: 

POSTULATE  M.  The  unaccelerated  motion  of  a  system  of 
reference  S  cannot  be  detected  by  observations  made  on  S  alone, 
the  units  of  measurement  being  those  belonging  to  S. 

The  postulate,  as  .stated,  is  a  direct  generalization  from 
experiment.  None  of  the  actually  existing  experimental  evidence 
is  opposed  to  it.  The  conviction  that  future  evidence  will 
continue  to  corroborate  it  is  so  strong  that  objection  has  seldom 
or  never  been  offered  to  this  postulate  by  either  the  friends 
or  the  foes  of  relativity.  No  means  at  present  known  will 
enable  the  observer  to  detect  motion  through  space  or  through 
any  sort  of  medium  which  may  be  supposed  to  pervade  space. 
Furthermore,  in  every  case  where  the  usual  theories  have 
predicted  the  possibility  of  detecting  such  motion  and  where 
sufficiently  exact  observations  have  been  made,  it  has  turned 
out  that  no  such  motion  was  detected.  Moreover,  one  at 
least  of  these  contradictions  of  theory — the  Michelson-Morley 
experiment — has  been  outstanding  for  a  period  of  twenty- 
five  years  and  no  satisfactory  explanation  has  been  offered  unless 
one  is  willing  to  accept  the  law  stated  in  postulate  M  above. 
It  would  appear,  therefore,  that  the  experimental  evidence  foi? 
the  postulate  is  to  be  considered  of  strong  character. 

§  11.  THE  SECOND  CHARACTERISTIC  POSTULATE. 

The  so-called  second  postulate  of  relativity,  in  the  form  in 
which  it  has  frequently  been  stated,*  involves  two  entirely 
distinct  parts.  To  the  present  writer  it  appears  that  no  incon- 
siderable part  of  the  difficulty  which  has  been  felt  concerning 

*  See  postulate  R  below  and  the  remarks  which  lead  up  to  it. 


18  THE   THEORY  OF   RELATIVITY. 

this  second  postulate  has  been  due  to  a  failure  to  perceive  the 
interdependence  of  these  two  parts  and  of  postulate  M  above. 
Precisely  that  part  of  the  second  postulate  to  which  most  objec- 
tion has  been  raised  is  a  logical  consequence  of  M  and  of  the 
other  part,  the  part  last  mentioned  being  a  statement  of  a 
law  which  for  a  long  time  has  been  accepted  by  physicists. 
Consequently,  we  shall  state  separately  the  two  parts  of  the 
second  postulate  and  bring  out  with  care  the  interdependence 
of  these  and  of  postulate  M  above. 

The  part  which  we  shall  give  first  states  a  principle  which 
has  long  been  familiar  in  the  theory  of  light,  namely,  that  the 
velocity  of  light  is  unaffected  by  the  velocity  of  the  source. 
Stated  in  exact  language  this  postulate  is  as  follows: 

POSTULATE  Rf .  The  velocity  of  light  in  free  space,  measured 
on  an  unaccelerated  system  of  reference  S  by  means  of  units  belong- 
ing to  S,  is  independent  of  the  unaccelerated  velocity  of  the  source 
of  light. 

The  law  stated  in  this  postulate  is  a  conclusion  which  follows 
readily  from  the  usual  undulatory  theory  of  light  and  will 
therefore  be  accepted  by  any  one  who  holds  to  that  theory. 
But  it  should  be  emphasized  that  R'  does  not  depend  for  its 
truth  on  any  theory  of  light.  It  is  a  matter  for  direct  experi- 
mental verification  or  disproof,  and  this  should  be  made  in  such 
a  way  as  to  be  independent,  as  far  as  possible,  of  all  general 
theories  of  light,  at  least  insofar  as  they  are  not  supported  by 
direct  experimental  evidence.  So  far  as  the  writer  is  aware, 
there  is  no  experimental  evidence  which  is  undoubtedly  opposed 
to  postulate  M ,  while  on  the  other  hand  there  is  direct  experi- 
mental evidence  which  is  believed  by  some  to  be  definitely 
in  its  favor.  Tolman,*  in  particular  has  considered  this  matter 
in  relation  to  the  Doppler  effect  and  to  the  velocity  of  light  from 
the  two  limbs  of  the  sun;  and  has  concluded  that  experiment 
bears  out  the  postulate.  Stewart, t  on  the  other  hand,  has 
examined  the  same  experiments  and  has  found  an  explanation 

*  Physical  Review,  31  (1910):  26-40. 
^Physical  Review,  32  (1911):  418-428. 


THE  POSTULATES  OF  RELATIVITY.  19 

for  them  in  Thomson's  electromagnetic  emission  theory  of  light. 
According  to  Stewart  these  experiments  are  in  agreement  with 
our  postulate  M  but  are  opposed  to  our  postulate  R' '.  All 
other  attempted  proof  or  disproof  of  the  postulate  appears 
to  be  in  the  same  state;  it  is  capable  of  two  interpretations 
which  are  directly  opposed  to  each  other  with  respect  to  their 
conclusions  as  to  the  validity  of  R'.  Thus  at  present  there 
is  no  undoubted  experimental  evidence  for  or  against  postu- 
late R'.  If  the  assumption  is  to  be  proved  at  all,  either  new 
experiments  must  be  devised  or  it  must  be  proved  by  indirect 
means  by  showing  that  it  is  a  consequence  of  experiment  and 
accepted  laws. 

Now  any  one  who  accepts  postulates  M  and  Rf  will  perforce 
accept  also  all  the  logical  consequences  which  necessarily  flow 
from  them.  Of  these  logical  consequences  we  shall  now  prove 
one  which  is  of  great  importance  in  the  theory  of  relativity: 

THEOREM  I.  The  velocity  of  light  in  free  space,  measured  on 
an  unaccelerated  system  of  reference  S  by  means  of  units  belonging 
to  S,  is  independent  of  the  direction  of  motion  of  S(MR')* 

Since  by  R'  the  velocity  of  light  is  independent  of  that  of 
the  light-source  we  may  suppose  that  the  light-source  belongs 
to  the  system  of  reference  S.  Now  let  the  velocity  of  light, 
as  it  is  emitted  from  this  source  in  various  directions,  be  observed 
and  tabulated.  On  account  of  the  homogeneity  and  isotropy 
of  space  mere  direction  through  space  will  have  no  effect  on  these 
observed  velocities;  and  therefore  if  they  differ  at  all,  the 
difference  will  be  due  to  the  velocity  of  5.  Now  if  there  were 
a  difference  due  to  the  direction  of  motion  of  S  this  difference 
would  put  in  evidence  the  motion  of  S.  But  by  M  it  is  impos- 
sible to  detect  such  motion  in  this  way.  Hence  the  observed 
velocity  must  be  the  same  in  all  directions.  In  other  words, 
it  is  independent  of  the  direction  of  motion  of  5;  and  thus 
the  theorem  is  proved. 

It  is  clear,  however,  that  we  cannot  take  the  next  step  and 

*  Letters  attached  to  a  theorem  in  this  way  indicate  those  of  the  postulates 
on  which  the  theorem  depends. 


20  THE   THEORY   OF   RELATIVITY. 

prove  that  this  observed  velocity  of  light  is  independent  of  the 
numerical  value  of  the  velocity  of  S.  To  see  this  clearly,  let 
us  suppose  that  the  numerical  value  of  the  velocity  of  S  does 
effect  the  observed  velocity  of  light.  On  account  of  Rf  it  will 
have  the  same  effect  on  the  observed  velocity  of  light  whatever 
may  be  the  unaccelerated  motion  of  the  light-source.  Hence, 
from  all  possible  observations,  the  experimenter  will  have  only 
a  single  datum  from  which  to  determine  the  effect  of  one  phe- 
nomenon on  another;  namely,  a  datum  in  which  the  two  phe- 
nomena are  connected  in  a  certain  definite  way.  It  is  obvious 
then  that  he  cannot  determine  the  effect  of  one  of  the  phenomena 
on  the  other;  for  he  can  never  observe  the  one  without  the 
other  being  present  also  and  the  connection  which  exists  between 
them  is  always  the  same  however  he  may  vary  the  experiment. 
And  if  the  observer  cannot  determine  an  existing  effect  it  is 
clear  that  he  cannot  prove  the  absence  of  any  effect  whatever. 
But,  although  the  absence  of  this  effect  cannot  be  proved,  it  is 
probably  impossible  to  conceive  any  satisfactory  way  in  which 
it  could  be  present.  Physical  intuition  is  emphatic  in  asserting 
that  if  the  direction  of  the  velocity  of  5  has  no  effect  on  the 
observed  velocity  of  light  then  the  numerical  value  of  the  veloc- 
ity of  S  has  no  effect  on  such  observed  velocity.  But  this  does 
not  constitute  a  proof.  There  is  in  this,  however,  nothing  to 
invalidate  the  naturalness  of  the  assumption  of  such  independence 
of  the  two  velocities;  in  fact,  it  would  be  unscientific  to  make 
a  different  assumption  (which  would  necessarily  introduce 
greater  complications)  unless  we  were  forced  to  it  by  unques- 
tioned experimental  fact.  Accordingly,  we  shall  make  the 
assumption  and  shall  state  it  as  postulate  R": 

POSTULATE  R".  The  velocity  of  light  in  free  space,  measured 
on  an  unaccelerated  system  of  reference  S  by  means  of  units  belong- 
ing to  5,  is  independent  of  the  numerical  value  of  the  velocity  of  S. 

POSTULATE  R.  The  postulate  obtained  by  combining  R' 
and  R"  will,  for  convenience,  often  be  referred  to  as  postulate  R. 

Now  since  unaccelerated  velocity  is  completely  determined 
when  the  numerical  value  of  the  velocity  and  the  direction  of 


THE   POSTULATES   OF   RELATIVITY.  21 

the  motion  are  given  the  truth  of  the  following  theorem  is  an 
immediate  consequence  of  theorem  I  and  postulate  R" : 

THEOREM  II.  The  velocity  of  light  in  free  space,  measured 
on  an  unaccelerated  system  of  reference  S  by  means  of  units  belong- 
ing to  S,  is  independent  of  the  velocity  of  S  (MR) . 

The  second  postulate  of  relativity  has  usually  been  stated 
in  a  form  different  from  that  given  above  in  Rf  and  R"  or  R. 
In  fact,  the  truth  of  theorem  I  has  often  been  taken  as  part  of 
the  assumption  in  this  postulate,  notwithstanding  that  I  can 
be  derived  from  M  and  R' '.  Now,  it  is  precisely  the  assump- 
tion of  I  that  has  given  most  difficulty  to  some  persons.  It 
is  believed  that  a  part  of  this  difficulty  will  disappear  in  view 
of  the  fact  that  I  is  here  demonstrated  by  means  of  M  and  R' '. 

For  the  sake  of  convenience  in  future  discussion  one  of  the 
customary  formulations  of  the  second  postulate  is  appended 
here.  It  must  be  remembered,  however,  that  it  is  not  a  separate 
constituent  part  of  our  present  body  of  doctrine  but  is  already 
contained  in  M  and  R,  in  part  directly  and  in  part  as  a  nec- 
essary consequence  of  these  postulates. 

POSTULATE  R.  The  velocity  of  light  in  free  space,  measured 
on  an  unaccelerated  system  of  reference  S  by  means  of  units  belonging 
to  S,  is  independent  of  the  velocity  of  S  and  of  the  unaccelerated 
velocity  of  the  light-source. 

From  the  very  nature  of  the  postulate  R"  it  is  difficult 
to  obtain  direct  experimental  evidence  for  or  against  it.  It 
seems,  however,  as  we  have  previously  pointed  out,  that  one 
who  accepts  theorem  I  can  hardly  refuse  to  assume  R" '.  But 
theorem  I  is  a  logical  consequence  of  postulates  M  and  R' ', 
as  we  have  shown.  Moreover,  from  what  follows  it  will  be 
seen  that  we  have  occasion  to  make  no  further  assumptions 
which  can  in  any  way  run  counter  to  currently  accepted  notions. 
Consequently,  it  would  seem  that  the  experimental  evidence  for 
or  against  the  whole  theory  of  relativity  must  center  around 
postulates  M  and  R'.  We  have  already  given  some  account 
of  the  experimental  evidence  for  these  postulates.  In  connec- 
tion with  theorems  to  be  derived  later  further  reference  will 


22  THE   THEORY   OF   RELATIVITY. 

be  given  to  the  existing  experimental  evidence  and  some  other 
possible  lines  of  research  in  this  direction  will  be  pointed  out. 

It  is  generally  conceded  that  the  strange  conclusions  which 
are  obtained  in  the  theory  of  relativity  are  due  to  postulate 
R  (or  to  postulate  R  in  the  customary  formulation).  In 
view  of  theorem  I  above  and  the  discussion  of  its  consequences, 
it  is  now  clear  that  the  strangeness  in  the  conclusions  of 
relativity  is  due  to  that  part  of  R  which  is  contained  in  R' '. 
It  is  important  therefore  to  have  a  careful  analysis  of  this  pos- 
tulate and  especially  to  know  alternative  forms,  which,  in  view 
of  the  other  postulates,  are  logically  equivalent  to  it.  We 
shall  return  to  this  matter  in  Chapter  VI. 

§  12.    THE  POSTULATES  V  AND  L. 

It  has  been  customary  for  writers  on  relativity  to  state 
explicitly  only  the  postulates  M  and  R.  But  every  one,  as  a 
matter  of  fact,  has  made  further  assumptions  concerning  the 
relations  of  the  two  systems.  These  assumptions  in  some  form 
are  essential  to  the  initial  arguments  and  to  the  conclusions 
which  are  drawn  by  means  of  them.  To  the  present  writer 
it  seems  preferable  to  have  these  assumptions  explicitly  stated. 
Among  several  forms,  any  one  of  which  might  be  chosen,  there 
is  one  which  seems  to  be  decidedly  simpler  than  any  of  the 
others;  and  it  is  this  one  which  we  shall  employ  here.  We 
state  the  postulates  V  and  L  as  follows: 

POSTULATE  V.  If  the  velocity  of  a  system  of  reference  $2 
relative  to  a  system  of  reference  Si  is  measured  by  means  of  the 
units  belonging  to  Si  and  if  the  velocity  of  Si  relative  to  £2  is 
measured  by  means  of  the  units  belonging  to  62  the  two  results 
will  agree  in  numerical  value. 

This  velocity  we  shall  call  the  relative  velocity  of  the  two 
systems.  The  direction  line  of  this  velocity  will  be  called  the 
line  of  relative  motion  of  the  two  systems. 

POSTULATE  L.  If  two  systems  of  reference  Si  and  £2  move 
with  unaccelerated  relative  velocity  and  if  a  line  segment  I  is  per- 


THE   POSTULATES    OF   RELATIVITY.  2$ 

pendicular  to  the  line  of  relative  motion  of  Si  and  82  and  is  fixed 
to  one  of  these  systems,  then  the  length  of  I  measured  by  means 
of  the  units  belonging  to  Si  will  be  the  same  as  its  length  measured 
by  means  of  the  units  belonging  to  6*2. 

The  essential  content  of  these  two  postulates  may  be  stated 
in  simpler  terms  (but  less  accurately)  if  one  allows  the  explicit 
introduction  of  the  observer.  Thus  V  is  roughly  equivalent 
to  the  following  statement:  Two  observers  whose  relative  motion 
is  uniform  will  agree  in  their  measurement  of  that  uniform  relative 
motion.  As  an  approximate  equivalent  of  L  we  have:  Two 
observers  whose  relative  motion  is  uniform  will  agree  in  their 
measurement  of  length  in  a  line  perpendicular  to  their  line  of 
relative  motion. 

It  will  be  observed  that  these  two  postulates  are  nothing 
more  than  explicit  statements  of  notions  which  underlie  the 
classic  theories  of  mechanics.  The  first  is  assumed  in  suppos- 
ing that  there  exists  such  a  thing  as  the  relative  motion  of  two 
bodies  which  are  not  at  rest  relatively  to  each  other.  The  second 
is  nothing  more  than  the  statement  of  a  portion  of  the  idea  which 
lies  at  the  bottom  of  our  conception  of  such  a  thing  as  the 
length  of  a  rod  or  other  object. 

Since  these  two  postulates  are  universally  accepted,  the 
question  might  naturally  arise,  Why  state  them  at  all?  Is 
it  not  enough  simply  to  take  them  for  granted?  The  answer 
is  that  there  are  other  notions  which  have  heretofore  met 
with  the  same  universal  acceptance  and  which  do  not  agree 
with  the  postulates  of  relativity.  Therefore  it  seems  to  be 
desirable — in  fact,  to  be  essential  to  proper  logical  procedure — 
to  state  explicitly  just  those  assumptions  concerning  the  rela- 
tion of  the  two  systems  of  reference  which  we  shall  have  occasion 
to  employ  in  argument.  Only  in  this  way  is  one  able  to  see 
exactly  on  what  basis  our  strange  conclusions  rest. 

We  shall  make  a  digression  here  to  say  one  further  word 
about  postulate  L.  In  the  next  chapter  we  shall  draw  the 
conclusion  that  length  in  the  line  of  motion  is  not  independent 
of  the  velocity  with  which  the  system  is  moving.  In  view  of 


24  THE   THEORY  OF  RELATIVITY. 

this  the  question  arises  as  to  why  we  must  assume  that  length 
in  a  line  perpendicular  to  the  line  of  motion  is  independent 
of  the  motion.  The  answer  is  that  we  are  under  no  such  ne- 
cessity, that  we  are  at  liberty  to  assume  that  length  in  a  line 
perpendicular  to  the  line  of  motion  is  dependent  on  the  velocity 
of  such  motion.  In  fact,  the  general  formulation  of  such  an 
hypothesis  has  already  been  made  by  E.  Riecke.*  This  hypoth- 
esis, however,  is  undoubtedly  more  complicated  and  less 
elegant  than  the  one  which  we  have  made;  and  the  latter,  as 
we  shall  see,  is  in  conflict  with  no  known  experimental  facts. 
Therefore,  following  that  instinct  which  has  always  wisely 
guided  the  physicist,  we  make  the  simplest  hypothesis  which 
is  in  agreement  with  and  explanatory  of  the  totality  of  exper- 
imental facts  at  present  known.  If  at  any  time  experiments 
are  set  forth  which  do  not  agree  with  the  theory  developed  on 
the  basis  of  the  above  postulates,  then  will  be  the  time  to  con- 
sider the  question  of  introducing  a  more  complicated  postulate 
in  place  of  our  postulate  L  above. 

§  13.    CONSISTENCY  AND  INDEPENDENCE  OF  THE 
POSTULATES. 

Throughout  our  treatment  it  will  be  assumed  that  the 
postulates  as  stated  above  are  consistent;  that  is  to  say,  no 
attempt  will  be  made  to  prove  their  consistency.  The  fact 
that  no  contradictory  conclusions  have  been  drawn  from  them 
will  be  accepted  as  (partial)  evidence  that  they  are  mutually 
consistent.  Moreover,  from  their  very  nature  and  from  the 
differing  range  of  applicability  of  the  several  postulates  it  is 
difficult  to  conceive  how  any  of  them  can  possibly  contradict 
conclusions  which  may  be  drawn  from  the  others. 

There  is  another  question  also  which  it  is  our  purpose 
to  pass  over  without  discussion,  namely,  the  question  of  the 
logical  independence  of  the  postulates.  Is  any  postulate  or  a 
part  of  any  postulate  a  logical  consequence  of  the  remaining 
postulates?  This  question  is  important  from  the  point  of  view 

*  Gottinger  Nachrichten,  Math.  Phys.,  1911,  pp.  271-277. 


THE   POSTULATES   OF   RELATIVITY.  25 

of  formal  logic,  but  in  the  present  case  its  value  to  physical 
science  is  probably  small. 

§  14.    OTHER  POSTULATES  NEEDED. 

From  the  postulates  stated  above  it  is  possible  to  draw  only 
those  conclusions  of  the  theory  of  relativity  which  are  of  a 
general  nature  and  have  to  do  merely  with  the  measurement 
of  time  and  space.  They  alone  are  employed  in  Chapters 
III  and  IV. 

If  it  is  desired  to  study  the  nature  of  mass  or  the  relation 
of  mass  and  energy  in  the  theory  of  relativity,  it  is  necessary 
to  have  some  assumption  concerning  mass  in  the  first  case 
and  concerning  both  mass  and  energy  in  the  second  case.  Thus 
we  might  assume  the  conservation  laws  of  energy,  electricity 
and  momentum  and  deduce  the  joint  consequences  of  these 
assumptions  and  those  given  above.  It  is  our  purpose  to  take 
up  these  matters  in  Chapters  V  and  VI.  It  is  convenient 
to  state  the  postulates  here;  and  this  we  do,  after  giving  some 
necessary  definitions. 

If  m,  M  and  v  are  respectively  the  mass,  momentum  and 
velocity  of  a  body  we  shall  assume  (as  in  the  classical  mechanics) 
that  they  are  connected  by  a  relation  of  the  form 

M  =  mv. 

We  shall  take  mass  and  velocity  to  be  the  fundamental  quanti- 
ties and  shall  define  momentum  in  terms  of  them  by  the  above 
relation. 

Likewise  we  shall  define  the  kinetic  energy  E  of  a  moving 
body  by  means  of  the  usual  relation 

E  =  J0  M dv = jjmvdv. 

Later  we  shall  see  that  "  mass  "  is  variable  and  is  not  in  general 
independent  of  the  direction  in  which  it  is  measured;  conse- 
quently, we  must  take  for  m  in  the  above  formulas  the  mass 
of  the  body  in  the  direction  of  its  motion. 

We  shall  take  for  granted  the  following  laws  of  conservation 
of  momentum  and  energy  and  electricity: 


26  THE   THEORY  OF  RELATIVITY. 

POSTULATE  C\.  The  sum  total  of  momentum  in  any  isolated 
system  remains  unaltered,  whatever  changes  may  take  place  in  the 
system,  provided  that  it  is  not  affected  by  any  forces  from  without. 

POSTULATE  €2-  The  sum  total  of  energy  in  any  isolated 
system  remains  unaltered,  whatever  changes  may  take  place  in 
the  system,  provided  that  it  is  not  affected  by  any  forces  from 
without. 

POSTULATE  €3.  The  sum  total  of  electricity  in  any  isolated 
system  remains  unaltered,  whatever  changes  may  take  place  in  the 
system,  provided  that  the  system  as  a  whole  neither  receives  elec- 
tricity from  nor  gives  out  electricity  to  bodies  not  belonging  to  the 
system. 

The  "  action  "  of  a  moving  body  in  passing  from  one  posi- 
tion to  another  may  be  denned  as  the  space  integral  of  the 
momentum  taken  over  the  path  of  motion.  If  we  denote  this 
action  by  A  we  have  therefore 

A  =  J  M  ds  =  j  mvds. 
Now  ds  =  vdt,  so  that  we  have  also 

A  =Jmv2dt. 
If  several  bodies  are  involved  we  have 


where  the  summation  is  for  the  various  bodies  in  the  system. 

We  may  state  the  fundamental  principle  of  least  action  in 
the  following  form: 

PRINCIPLE  OF  LEAST  ACTION.  The  free  motion  of  a  con- 
servative system  between  any  two  given  configurations  has  the 
property  that  the  action  A  is  a  minimum,  the  admissible  values 
A  of  the  action  with  which  A  is  compared  being  obtained  from 
varied  motions  in  which  the  total  energy  has  the  same  constant 
value  as  in  the  actual  free  motion. 


CHAPTER  III. 
THE  MEASUREMENT  OF  LENGTH  AND   TIME. 

§  15.  RELATIONS  BETWEEN  THE  TIME  UNITS  OF  Two  SYSTEMS. 

LET  us  consider  three  systems  of  reference  S,  Si  and  £2 
related  to  each  other  in  the  following  manner:  The  lines  of 
relative  motion  of  S  and  Si,  of  S  and  £2,  of  Si  and  82  are  all 
parallel;  6*1  and  £2  have  a  relative  velocity*  v;  S  and  Si 
have  a  relative  velocity  \D  in  one  sense  and  S  and  £2  have  a 
relative  velocity  %v  in  the  opposite  sense.  The  system  S  con- 
sists of  a  single  light-source,  and  this  source  is  symmetrically 
placed  with  respect  to  two  points  of  which  one  is  fixed  to  Si 
and  the  other  is  fixed  to  £2.  This  is  possible  as  a  permanent 
relation  on  account  of  the  relative  motions  of  the  three  systems. 
For  convenience,  let  us  assume  £  to  be  at  rest. 

We  shall  now  suppose  that  observers  on  the  systems  £1  and 
£2  measure  the  velocity  of  light  as  it  emanates  from  the  source  £. 
Let  a  point  A  on  Si  and  a 
point   B   on   £2,   which    are         i* 
symmetrically     placed     with 
respect  to  the  light-source  £, 


move  along  the  lines  h  and        ~^      ~A~  D    ~E 


fa;   these   lines  are    parallel.  FlG  3 

From  postulate  L  it  follows 

that  the  observers  on  £1  and  £2  will  obtain  the  same  measure- 
ment of  the  distance  between  l\  and  fa.  Denote  this  distance 
by  J.  From  postulate  M  it  follows  that  neither  observer  is 
able  to  detect  his  motion.  Therefore  he  will  make  his  observa- 
tions on  the  assumption  that  his  system  is  at  rest;  that  is  to 
say,  his  measurements  will  be  made  by  means  of  the  units 

*  Note  that  postulate  V  is  required  to  make  this  hypothesis  legitimate. 

27 


28  THE   THEORY   OF   RELATIVITY. 

belonging  to  his  system  and  no  corrections  will  be  made  on 
account  of  the  motion  of  the  system.  Let  the  observer  on  Si 
reflect  a  ray  of  light  5^4  from  a  point  A  to  a  point  C  on  fa  and 
back  to  A;  and  let  the  observed  time  of  passage  of  the  light 
from  A  to  C  and  back  to  A  be  /.  Since  the  observer  assumes 
his  system  to  be  at  rest  he  will  suppose  that  the  ray  of  light 
passes  (in  both  directions)  along  the  line  AC  which  is  perpendic- 
ular to  h  and  fa.  His  measurement  of  the  distance  traversed 
by  the  ray  of  light  in  time  /  will  therefore  be  2d.  Hence  he 
will  obtain  as  a  result 


where  c  is  his  observed  velocity  of  light. 

Similarly,  an  observer  on  62,  supposing  his  system  to  be  at 
rest  finds  the  time  t\  which  it  requires  for  a  ray  of  light  to  pass 
from  B  to  D  and  return,  the  ray  employed  being  gotten  by 
reflecting  a  ray  SB  at  B.  Thus  the  second  observer  obtains 

2d 

Ti=C^ 
where  c\  is  his  observed  velocity  of  light. 

Now,  from  the  assumed  relations  among  the  systems  Sy 
Si  and  ^2  and  from  the  homogeneity  of  space  it  follows  that  the 
two  observations  which^we  have  supposed  to  be  made  must  lead 
to  the  same  estimate  for  the  velocity  of  light.  This  is  readily 
seen  from  the  fact  that  the  observations  were  made  in  such  a 
way  that  the  effect  due  to  either  the  numerical  value  or  the 
direction  of  the  motion  of  the  systems  Si  and  £2  is  the  same 
in  the  two  cases.  In  other  words,  if  we  denote  by  LI  and  Lz 
the  quantities  measured  on  Si  and  S2  respectively,  then  the 
relation  of  L\  to  Si  is  precisely  the  same  as  that  of  L*  to  Sz  ; 
and  hence  the  numerical  results  are  equal,  as  one  sees  from  the 
definition  of  systems  of  reference.  Therefore  we  have  ci=c. 

Let  us  now  suppose  that  the  observer  at  A  is  watching  the 
experiment  at  B.  To  him  it  appears  that  B  is  moving  with  a 
velocity  v,  since  by  hypothesis  the  two  systems  have  the  relative 
velocity  v  and  A  and  B  measure  this  velocity  alike.  We  shall 


THE  MEASUREMENT  OF  LENGTH  AND  TIME.  29 

assume  that  the  apparent  motion  is  in  the  direction  indicated 
by  the  arrow  in  the  figure.  To  the  observer  at  B  it  appears 
that  the  ray  of  light  traverses  BD  from  B  to  D  and  returns 
along  the  same  line  to  B.  To  the  observer  at  A  it  appears 
that  the  ray  traverses  the  line  BEF,  F  being  the  point  which 
B  has  reached  by  the  time  that  the  ray  has  returned  to  the 
observer  at  this  point.  If  EG  is  perpendicular  to  lz  and  d\ 
is  the  length  of  EF  as  measured  by  means  of  units  belonging 
to  Sij  then,  evidently,  GF  (when  measured  in  the  same  units) 
is  grfi,  where  $=v/c  and  c  is  the  (apparent)  velocity  of  light  as 
estimated  in  this  case  by  the  observer  at  A.  From  the  right 
triangle  EFG  it  follows  at  once  that  we  have 


Now,  if  I  is  the  time  which  is  required,  according  to  the  observer 
at  A,  for  the  light  to  traverse  the  path  BEF,  then  we  have 


r=,-^==c. 


So  far  in  our  argument  in  this  section  we  have  employed 
only  those  of  our  postulates  which  are  generally  accepted  by 
both  the  friends  and  the  foes  of  relativity.  Now  we  come  to 
the  place  where  the  men  of  the  two  camps  must  part  company. 

Let  us  introduce  for  the  moment  the  following  additional 
hypothesis: 

.ASSUMPTION  A.  The  two  estimates  c  and  c  of  the  velocity 
of  light  obtained  as  above  by  the  observer  at  A  are  equal. 

Now  we  have  shown  that  c  is  equal  to  c\.  Hence  we  may 
equate  the  values  of  a  and  ~c  given  above;  thus  we  have 


or 


But  ti  and  t  are  measures  of  the  same  interval  of  time,  t\  being 
in  units  belonging  to  52  and  I  being  in  units  belonging  to  Si. 
Hence  to  the  observer  on  Si,  the  ratio  of  his  time  unit  to  that  of 


30  THE   THEORY  OF  RELATIVITY. 


the  system  62  appears  to  be  Vi-g2  :  i.  On  the  other  hand, 
it  may  be  shown  in  exactly  the  same  way  that  to  the  observer 
on  52  the  ratio  of  his  time  unit  to  that  of  the  system  Si  appears 
to  be  Vi  —  @2  :  i.  That  is,  the  time  units  of  the  two  systems 
are  different  and  each  observer  comes  to  the  same  conclusion 
as  to  the  relation  which  the  unit  of  the  other  system  bears  to 
his  own. 

This  important  and  striking  result  may  be  stated  in  the 
following  theorem: 

THEOREM  III.  If  two  systems  of  reference  Si  and  £2  move 
with  a  relative  velocity  v  and  g  is  defined  as  the  ratio  of  v  to  the 
velocity  of  light  estimated  in  the  manner  indicated  above,  then  to 
an  observer  on  Si  the  time  unit  of  Si  appears  to  be  in  the  ratio 
Vi  —  $2  :  i  to  that  of  S2  while  to  an  observer  on  82  the  time  unit 
oj  S2  appears  to  be  in  the  ratio  Vi  -  $2  :  i  to  that  of  Si  (MVLA). 

Let  us  now  bring  into  play  our  postulate  R' '.  In  theorem  I 
we  have  already  seen  that  a  logical  consequence  of  M  and  R' 
is  that  the  velocity  of  light,  as  observed  on  a  system  of  reference, 
is  independent  of  the  direction  of  motion  of  that  system.  Now, 
if  c  and  c  as  estimated  above  differ  at  all,  that  difference  can  be 
due  only  to  the  direction  of  motion  of  Si,  as  one  sees  readily 
from  postulate  Rf  and  the  method  of  determining  these  quan- 
tities. Hence  the  statement  which  we  made  above  as  assump- 
tion A  is  a  logical  consequence  of  postulates  M  and  R' .  There- 
fore we  are  led  to  the  following  corollary  of  the  above  theorem: 

COROLLARY.  Theorem  III  may  be  stated  as  depending  on 
(MVLRf)  instead  of  on  (MVLA). 

Let  us  now  go  a  step  further  and  employ  postulate  R". 
From  theorem  I  and  postulates  R'  and  R"  it  follows  that  the 
observed  velocity  of  light  is  a  pure  constant  for  all  admissible 
methods  of  observation.  If  we  make  use  of  this  fact  the 
preceding  result  may  be  stated  in  the  following  simpler  form: 

THEOREM  IV.  If  two  systems  of  reference  Si  and  S2  move 
with  a  relative  velocity  v  and  $  is  the  ratio  of  v  to  the  velocity  of  light, 
then  to  an  observer  on  Si  the  time  unit  of  Si  appears  to  be  in  the 
ratio  Vi  —  $2  :  i  to  that  of  S2  while  to  an  observer  on  Sz  the  time 


THE  MEASUREMENT  OF  LENGTH  AND  TIME.  31 


unit  of  $2  appears  to  be  in  the  ratio  Vi  —  @2  :  i  to  that  of  Si 
(MVLR). 

Let  us  subject  these  remarkable  results  to  a  further 
analysis.  Theorem  III,  its  corollary  and  theorem  IV  all  agree 
in  the  extraordinary  conclusion  that  the  time  units  of  the  two 
systems  of  reference  Si  and  52  are  of  different  lengths.  Just 
how  much  they  differ  is  a  secondary  matter;  that  they  differ 
at  all  is  the  surprising  and  important  thing.  As  postulates 
M,  V,  L  are  generally  accepted  and  have  not  elsewhere  led 
to  such  strange  conclusions  it  is  natural  to  suppose  that  the 
strangeness  here  is  not  due  to  them. 

Referring  to  the  argument  carried  out  above,  we  see  that 
no  unusual  conclusions  were  reached  until  we  had  introduced 
and  made  use  of  assumption  A.  Moreover  we  have  seen 
that  this  assumption  itself  is  a  logical  consequence  of  M  and  R'. 
Further,  R"  is  not  involved  either  in  theorem  III  or  in  its 
corollary.  But  these  already  contain  the  strange  features  of 
our  results.  Hence  the  conclusion  is  irresistible  that  the 
extraordinary  element  in  these  results  is  due  to  postulate  Rf — 
or  to  speak  more  accurately,  to  just  that  part  of  it  which  it  is 
necessary  to  use  in  connection  with  M  in  order  to  prove  A  as 
a  theorem. 

This  result  is  important,  as  the  following  considerations 
show.  Postulates  V  and  L  state  laws  which  have  been  univer- 
sally accepted  in  the  classical  mechanics.  Postulate  M  is  a 
direct  generalization  from  experiment,  and  the  generalization 
is  legitimate  according  to  the  usual  procedure  of  physicists 
in  like  situations.  Postulate  Rf  is  a  statement  of  a  principle 
which  has  long  been  familiar  in  the  theory  of  light  and  has 
met  with  wide  acceptance.  Thus  we  see  that  no  one  of  these 
postulates,  in  itself,  runs  counter  to  currently  accepted  physical 
notions.  And  yet  just  these  postulates  alone  are  sufficient 
to  enable  us  to  conclude  that  corresponding  time  units  in 
two  systems  of  reference  are  of  different  magnitude.  In  the 
next  section  we  shall  show  on  the  basis  of  the  same  postulates 
that  the  corresponding  units  of  length  in  the  two  systems 
are  also  different.  Thus  the  most  remarkable  elements  in  the 


32  THE  THEORY  OF  RELATIVITY. 

conclusions  of  the  theory  of  relativity  are  deducible  from 
postulates  M,  V,  L,  R'  alone;  and  yet  these  are  either  generaliza- 
tions from  experiment  or  statements  of  laws  which  have  usually 
been  accepted.  Hence  we  conclude:  The  theory  of  relativity, 
in  its  most  characteristic  elements,  is  a  logical  consequence  of 
certain  generalizations  from  experiment  together  with  certain 
laws  which  have  for  a  long  time  been  accepted. 

One  other,  remark,  of  a  totally  different  nature,  should  be 
made  with  reference  to  the  characteristic  result  of  theorem 
IV.  It  has  to  do  with  the  relation  between  the  time  units  of 
the  two  systems.  This  relation  is  intimately  associated  with 
the  fact  that  each  observer  makes  his  measurements  on  the 
hypothesis  that  his  own  system  is  at  rest,  while  the  other  sys- 
tem is  moving  past  him  with  the  velocity  v.  If  both  observers 
should  agree  to  call  S  fixed  and  if  further  in  this  modified 
"  universe  "  our  postulates  V,  L,  R  were  still  valid  it  would 
turn  out  that  the  two  observers  would  find  their  time  units 
in  agreement.  But,  in  view  if  M,  the  choice  of  5  as  fixed  would 
undoubtedly  seem  perfectly  arbitrary  to  both  observers;  and 
the  content  of  the  modified  postulate  R  would  be  essentially 
different  from  that  of  the  postulate  as  we  have  employed  it. 
Hence,  if  we  accept  R  as  it  stands — or,  indeed,  even  a  certain 
part  of  it,  as  we  have  shown  above — we  must  conclude  that 
the  time  units  in  the  two  systems  are  not  in  agreement,  in  fact, 
that  their  ratio  is  that  stated  in  the  theorems  above. 

§  16.  RELATIONS  BETWEEN  THE  UNITS  OF  LENGTH  OF  Two 

SYSTEMS. 

Let  us  consider  three  systems  of  reference  S,  Si  and  62 
related  in  the  same  manner  as  in  the  preceding  section  except 
that  now  the  two  lines  h  and  h  coincide.  We  suppose  that  S\ 
is  moving  in  the  direction  indicated  by  the  arrow  at  A  and  that 
52  is  moving  in  the  direction  indicated  by  the  arrow  at  B. 

We  suppose  that  observers  at  A  and  B  again  measure  the 
velocity  of  light  as  it  emanates  from  51,  this  time  in  the  direc- 
tion of  the  line  of  motion.  Each  will  carry  out  his  observations 


THE  MEASUREMENT  OF  LENGTH  AND  TIME.  33 

on  the  supposition  that  his  system  is  at  rest,  for  from  M  it  fol- 

lows that  he  cannot  detect  the  motion  of  his  system.     The 

observer  at  A  measures  the 

time  ti  of  passage  of  a  ray  of          Zl     ^  _  _    p      la 

light  from  A  to  C  and  return  *^i        s        £~ 

to  A,  the  length  of  AC  being  FIG.  4. 

d  when  the   measurement  is 

made  with  a  unit  belonging  to  Si.    Likewise,  the  observer  at 

B  measures  the  time  /2  of  passage  of  a  ray  of  light  from  B  to 

D  and  return  to  B,  the  length  BD  being  d  when  measured  with 

a  unit  belonging  to  S%. 

Just  as  in  the  preceding  case  it  may  be  shown  that  the  two 
observers  must  obtain  the  same  estimate  for  the  velocity  of 
light.  But  the  estimate  of  the  observer  at  A  is  2d/t\  while 
that  of  the  observer  at  B  is  2d/t2.  Hence 


that  is,  the  number  of  units  of  time  required  for  the  passage  of 
the  ray  at  A  and  of  the  ray  at  B  is  the  same,  the  former  being 
measured  on  Si  and  the  latter  on  52-  Moreover,  the  measure 
of  length  is  the  same  in  the  two  cases.  But  the  units  of  time, 
as  we  saw  in  the  preceding  section,  do  not  have  the  same  magni- 
tude. Hence  the  units  of  length  of  the  two  systems  along 
their  line  of  motion  do  not  have  the  same  magnitude;  and  the 
ratio  of  units  of  length  is  the  same  as  the  ratio  of  units  of  time. 

Combining  this  result  with  theorem  III,  its  corollary  and 
theorem  IV  we  have  the  following  three  results: 

THEOREM  V.  If  two  systems  of  reference  Si  and  5*2  move 
with  a  relative  velocity  v  and  &  is  defined  as  the  ratio  of  v  to  the 
velocity  of  light  estimated  in  the  manner  indicated  in  the  first 
part  of  §15,  then  to  an  observer  on  Si  the  unit  of  length  of  Si  along 
the  line  of  relative  motion  appears  to  be  in  the  ratio  Vi  —  g2  :  i 
to  that  of  $2  while  to  an  observer  on  62  the  unit  of  length  of  62 
along  the  line  of  relative  motion  appears  to  be  in  the  ratio  Vi  —  $2  :  i 
to  that  of  Si(MVLA). 

COROLLARY.  Theorem  V  may  be  stated  as  depending  on 
(MVLR')  instead  of  on  (MVLA). 


34  THE   THEORY   OF   RELATIVITY. 

THEOREM  VI.  If  two  systems  of  reference  S\  and  82  move 
with  a  relative  velocity  v  and  g  is  the  ratio  of  v  to  the  velocity  of  light, 
then  to  an  observer  on  S\  the  unit  of  length  of  Si  along  the  line  of 
relative  motion  appears  to  be  in  the  ratio  Vi  —  (32  :  i  to  that  of  82 
while  to  an  observer  on  82  the  unit  of  length  of  8-2  along  the  line  of 
relative  motion  appears  to  be  in  the  ratio  Vi  —  £2  :  i  to  that  of  Si 
(MVLR). 

We  might  make  an  analysis  of  these  results  similar  to  that 
which  we  gave  for  the  corresponding  results  in  the  preceding 
section.  But  it  would  be  largely  a  repetition.  It  is  sufficient 
to  point  out  that  the  remarkable  conclusions  as  to  units  of  length 
in  the  two  systems  rest  on  just  those  postulates  which  led  to 
the  strange  results  as  to  the  units  of  time. 

§  17.  DISCUSSION  OF  THE  NOTION  OF  LENGTH. 

In  the  preceding  section  we  saw  that  two  observers  A  and  B 
on  relatively  moving  systems  of  reference  Si  and  £2  respectively 
are  in  a  very  peculiar  disagreement  as  to  units  of  length  along 
a  line  /  parallel  to  their  line  of  relative  motion.  To  A  it  appears 
that  £'s  units  are  longer  than  his  own.  On  the  other  hand, 
it  seems  to  B  that  his  units  are  shorter  than  A's.  In  the  two 
cases  the  apparent  ratio  is  the  same;  more  precisely,  the  unit 
which  appears  to  either  observer  to  be  the  shorter  seems  to 
him  to  have  the  ratio  Vi  —  £2  :  i  to  that  which  appears  to  him 
to  be  the  longer.  Although  they  are  thus  in  disagreement 
there  is  yet  a  certain  symmetry  in  the  way  in  which  their 
opinions  diverge. 

Let  us  suppose  that  these  two  observers  now  undertake  to 
bring  themselves  into  a  closer  agreement  in  measurements  of 
length  along  the  line  /.  Suppose  that  B  agrees  arbitrarily  to 
shorten  his  unit  so  that  it  will  appear  to  A  that  the  units  of  A 
and  B  are  of  the  same  length.  Then,  so  far  as  A  is  concerned, 
all  difficulty  has  disappeared.  How  is  B  affected  by  this  change? 
We  see  that  the  difficulty  which  he  experienced  is  not  disposed 
of;  on  the  other  hand  it  is  greater  than  before.  Already,  it 
seemed  to  him  that  his  unit  was  shorter  than  ^4's.  Now,  since 


THE   MEASUREMENT   OF   LENGTH   AND   TIME.  .    35 

he  has  shortened  his  unit,  the  divergence  appears  to  him  to  be 
increased.  Moreover,  the  symmetry  which  we  found  in  the 
former  case  is  now  absent. 

Furthermore,  if  any  other  changes  in  the  units  of  A  and  B 
are  made  we  shall  always  find  difficulties  as  great  as  or  greater 
than  those  which  we  encountered  in  the  initial  case.  There  is 
no  other  conclusion  than  this :  We  are  face  to  face  with  an  essen- 
tial difficulty — one  that  is  not  to  be  removed  by  any  mere 
artifice.  What  account  of  it  shall  we  render  to  ourselves? 

This  much  is  already  obvious:  The  length  of  an  object 
depends  in  an  essential  way  upon  the  measurer  and  the  system 
to  which  he  belongs. 

We  have  certain  intuitive  notions  concerning  the  nature  of 
matter  which  it  is  necessary  for  us  to  examine  if  we  are  to  dis- 
cuss adequately  the  notion  of  length.  We  have  usually  supposed 
that  to  revolve  a  steel  bar,  for  instance,  through  an  angle  of 
ninety  degrees  has  no  effect  upon  its  length.  Let  us  suppose 
for  the  moment  that  this  is  not  so;  but  that  the  bar  is  shorter 
when  pointing  in  some  directions  than  in  others,  so  that  its  length 
is  the  product  of  two  factors  one  of  which  is  its  length  in  a  cer- 
tain initial  position  and  the  other  of  which  is  a  function  of  the 
direction  in  which  the  body  points  relative  to  that  in  the  initial 
position.  Suppose  that  at  the  same  time  all  other  objects 
experience  precisely  the  same  change  for  varying  directions. 
It  is  obvious  that  in  this  case  we  should  have  no  means  of 
ascertaining  this  dependence  of  length  upon  the  direction  in 
which  the  body  points. 

To  an  observer  placed  in  a  situation  like  this  it  would  be 
natural  to  assume  that  the  length  of  the  steel  bar  is  the  same 
in  all  directions.  In  other  words,  in  arriving  at  his  definition 
of  length  he  would  make  certain  conventions  to  suit  his  con- 
venience. 

Now  suppose  that  the  system  of  such  an  observer  is  set  in 
motion  with  a  uniform  velocity  v  relative  to  the  previous  state 
of  the  system;  and  that  at  the  same  time  all  bodies  on  his 
system  undergo  simultaneously  a  continuous  dilatation  or  contrac- 


36  THE   THEORY   OF   RELATIVITY. 

tion.  This  observer  would  have  no  means  of  ascertaining  that 
fact;  and  accordingly  he  would  suppose  that  his  steel  bar 
had  the  same  length  as  before.  In  other  words,  he  would 
unconsciously  introduce  a  new  convention  concerning  his 
measurement  of  length. 

There  is  no  a  priori  reason  why  our  actual  universe  should 
not  be  such  as  the  hypothetical  one  just  described.  To  sup- 
pose it  so  unless  our  experience  demands  such  a  supposition 
would  be  unnatural;  because  it  would  introduce  an  unnecessary 
inconvenience.  But  suppose  that  in  our  growing  knowledge  of  the 
universe  there  should  come  a  time  when  we  could  more  con- 
veniently represent  to  ourselves  the  actual  facts  of  experience  by 
supposing  that  all  material  things  are  subject  to  some  such 
deformations  as  those  which  we  have  indicated  above;  there  is 
certainly  no  a  priori  reason  why  we  should  not  conclude  that 
such  is  the  essential  nature  of  the  structure  of  the  universe. 

Naturally  we  would  not  come  to  this  conclusion  without 
due  consideration.  We  would  first  enquire  carefully  if  there 
is  not  some  more  convenient  way  by  which  we  can  reconcile 
all  experimental  facts;  and  only  in  the  event  of  a  failure  to  find 
such  a  way  would  we  be  willing  to  modify  so  profoundly  our 
views  of  the  material  world. 

Now,  if  we  agree  to  suppose  that  our  actual  universe  is 
subject  to  a  certain  (appropriately  defined)  deformation  of 
the  general  type  discussed  above  it  would  follow  that  observers 
A  and  B  on  the  respective  systems  Si  and  £2  would  be  in  a  dis- 
agreement as  to  units  of  length  similar  to  that  which  exists, 
according  to  the  theory  of  relativity.  Therefore,  that  which  at 
the  outset  seemed  to  be  of  such  essential  difficulty  is  explained 
easily  enough,  if  we  are  willing  to  modify  so  profoundly  our 
conception  of  the  nature  of  material  bodies. 

Whether  in  the  present  state  of  science  experimental  facts 
demand  such  a  radical  procedure  is  a  question  which  will 
be  answered  differently  by  different  minds.  To  one  who 
accepts  the  postulates  of  relativity  there  is  indeed  no  other 
recourse;  one  who  refuses  to  accept  them  must  find  some  other 


THE  MEASUREMENT  OF  LENGTH  AND  TIME.  37 

satisfactory  way  to  account  for  experimental  facts.  The 
Lorentz  theory  of  electrons  gives  striking  evidence  in  favor 
of  supposing  that  matter  is  subject  to  some  such  deforma- 
tions as  those  mentioned  above;  and  this  evidence  is  the  more 
important  and  interesting  in  that  the  deformations  (as  con- 
ceived in  this  theory)  were  assumed  to  exist  simply  in  order 
to  be  able  to  account  directly  for  experimental  facts. 


§  18.    DISCUSSION  OF  THE  MEASUREMENT  OF  TIME.* 

That  two  observers  in  relative  motion  are  in  hopeless  dis- 
agreement as  to  the  measurement  of  length  in  their  line  of 
relative  motion  is  a  conclusion  which  is  probably  (at  first) 
sufficiently  disconcerting  to  most  of  us;  but  it  is  an  even  greater 
shock  to  intuition  to  conclude,  as  we  are  forced  to  do  accord- 
ing to  the  theory  of  relativity,  that  there  is  a  like  ineradicable 
disagreement  in  the  measurement  of  time.  A  discussion 
similar  to  that  in  the  preceding  section  brings  out  the  fact 
that  our  observers  A  and  B  cannot  possibly  arrive  at  con- 
sistent means  of  measuring  intervals  of  time.  The  treatment 
is  so  far  similar  to  the  preceding  discussion  for  length  that  we 
need  not  repeat  it;  we  shall  content  ourselves  with  a  brief 
discussion  of  conclusions  to  be  drawn  from  the  matter. 

Why  is  this  inability  of  A  and  B  to  agree  in  measuring  time 
received  in  our  minds  with  such  a  distinct  feeling  of  surprise 
and  shock?  It  is  doubtless  because  we  have  such  a  lively  sense 
of  the  passage  of  time.  It  seems  to  be  a  thing  which  we  know 
directly,  and  the  conclusion  in  question  is  contrary  to  our 
unsophisticated  intuition  concerning  the  nature  of  time. 

But  what  is  it  that  we  know  directly?  We  have  an  imme- 
diate perception  of  what  it  is  for  two  conscious  phenomena 
to  coexist  in  our  mind,  and  consequently  we  perceive  imme- 

*In  connection  with  this  section  and  the  following  one  the  reader  should 
compare  the  excellent  and  interesting  treatment  of  the  problem  of  measuring 
time  to  be  found  in  Chapter  II  of  Poincare's  Value  of  Science  (translated  into 
English  by  Halsted). 


38  THE   THEORY   OF   RELATIVITY. 

diately  the  simultaneity  of  events  in  our  mind.  Further,  we 
have  a  perfectly  clear  sense  of  the  order  of  succession  of  events 
in  our  own  consciousness.  Is  not  that  all  that  we  know  directly? 

The  difficulties  which  A  and  B  experience  in  correlating 
their  measurements  of  time  grow  out  of  two  things,  of  neither 
of  which  we  have  direct  perception. 

In  the  first  place  there  are  two  consciousnesses  involved; 
and  what  reason  have  we  to  suppose  that  succession  of  events 
is  the  same  for  these  two?  This  question  we  shall  not  treat, 
assuming  that  the  principal  matter  can  be  put  into  such  imper- 
sonal form  as  to  obviate  this  difficulty  altogether.  (As  a  mat- 
ter of  fact,  so  far  as  anything  characteristic  of  the  theory  of 
relativity  is  concerned  this  can  be  done.) 

The  other  difficulty  has  to  do  with  the  measurement  of  time 
as  opposed  to  the  mere  psychological  experience  of  its  passage. 
In  this  matter  we  are  entirely  without  any  direct  intuition  to 
guide  us.  We  have  no  immediate  sense  of  the  equality  of  two 
intervals  of  time.  Therefore,  whatever  definition  we  employ 
for  such  equality  will  necessarily  have  in  it  an  important  ele- 
ment of  convention.  To  keep  this  well  in  mind  will  facilitate 
our  discussion. 

Our  problem  is  this:  How  shall  we  assign  a  numerical 
measure  of  length  to  a  given  time  interval;  say  to  an  interval 
in  which  a  given  physical  phenomenon  takes  place?  We 
shall  arrive  at  the  answer  by  asking  another  question:  Why 
should  we  seek  to  measure  time  intervals  at  all,  seeing  that 
we  have  no  immediate  consciousness  of  the  equality  of  such 
intervals?  There  can  be  only  one  answer:  we  seek  to  measure 
time  as  a  matter  of  convenience  to  us  in  representing  to  our- 
selves our  experiences  and  the  phenomena  of  which  we  are 
witnesses.  In  such  a  way  we  can  render  to  ourselves  a  better 
account  of  the  world  in  which  we  live  and  of  our  relation  to  it. 

Now,  since  our  only  reason  for  attempting  to  measure  time 
is  in  a  matter  of  convenience,  the  way  in  which  we  measure 
it  will  be  determined  by  the  dictates  of  that  convenience.  The 
system  of  time  measurement  which  we  shall  adopt  is  just  that 


THE  MEASUREMENT  OF  LENGTH  AND  TIME.  39 

system  by  means  of  which  the  laws  of  nature  may  be  stated 
in  the  simplest  form  for  our  comprehension. 

Let  us  return  to  the  case  of  the  two  observers  A  and  B 
of  the  preceding  section.  Suppose  that  each  of  them  has  chosen 
a  system  of  measuring  time  that  suits  his  convenience  in  the 
interpretation  of  the  laws  of  nature  on  his  system.  There  is 
no  a  priori  reason  why  the  two  observers  should  measure  time 
intervals  in  the  same  way.  In  fact,  since  there  is  an  arbitrary 
element  in  the  case  of  each  method  of  measurement  and  since 
the  two  systems  are  in  a  state  of  relative  motion,  it  is  not  at 
all  unnatural  that  the  units  of  A  and  B  should  differ. 

Now  it  is  to  be  noticed  that  each  of  the  observers  A  and 
B  is  in  just  the  situation  in  which  we  find  ourselves.  We  have 
chosen  a  method  of  measuring  time  which  seems  to  us  conven- 
ient. Insofar  as  that  method  depends  on  convenience  it  is  rela- 
tive to  us  who  are  observers,  and  therefore  it  has  in  it  something 
which  is  arbitrary.  There  is  no  doubt  that  it  would  be  desir- 
able for  us  to  know  what  it  is  which  is  arbitrary,  which  is  relative 
to  us  who  observe;  but  it  is  equally  obvious  that  it  must  be 
difficult  for  us  to  determine  what  this  arbitrary  element  is. 

The  theory  of  relativity  makes  a  contribution  to  the  solu- 
tion of  this  problem.  We  suppose  that  two  observers  on  dif- 
ferent systems  find  the  laws  of  nature  the  same  as  we  find  them; 
or,  more  exactly,  we  suppose  that  they  find  certain  specific 
laws  the  same  as  we  find  them.  Then  we  inquire  as  to  their 
agreement  in  measuring  time  and  see  that  they  differ  in  a 
certain  definite  way.  This  difference  is  due  to  things  which 
are  relative  to  the  two  observers;  and  thus  we  begin  to  get 
some  insight  into  the  ultimate  basis  of  our  own  method  of 
measurement.  It  is  obviously  an  important  service  which 
the  theory  of  relativity  renders  to  us  when  it  enables  us  to 
to  make  an  advance  towards  a  better  understanding  of  such  a 
fundamental  matter  as  this. 

This  matter  will  become  clearer  if  we  speak  of  the  simulta- 
neity of  events  which  happen  at  different  places ;  and  therefore 
we  turn  to  a  discussion  of  this  topic. 


40  THE  THEORY  OF  RELATIVITY. 

§  19.  SIMULTANEITY  OF  EVENTS  HAPPENING  AT  DIFFERENT 

PLACES. 

Let  us  now  assume  two  systems  of  reference  5  and  S'  moving 
with  a  uniform  relative  velocity  v.  Let  an  observer  on  S' 
undertake  to  adjust  two  clocks  at  different  places  so  that  they 
shall  simultaneously  mark  the  same  hour.  We  will  suppose 
that  he  does  this  in  the  following  very  natural  manner:  Two 
stations  A  and  B  are  chosen  in  the  line  of  relative  motion  of  S 
and  S'  and  at  a  distance  d  apart.  The  point  C  midway  between 
these  two  stations  is  found  by  measurement.  The  observer 

is  himself  stationed   at  C  and 
has  assistants  at  A  and  B.     A 

A       TaCl  %d        B  t  ...  .       n 

c  *-*-         single  light  signal  is  flashed  from 

FIG.  5.  C  to  A  and  to  B,  and  as  soon 

as  the  light   ray  reaches  each 

station  the  clock  there  is  set  at  an  hour  agreed  upon  before- 
hand. The  observer  on  Sf  now  concludes  that  his  two  clocks, 
the  one  at  A  and  the  other  at  B,  are  simultaneously  marking 
the  same  hour;  for,  in  his  opinion  (since  he  supposes  his 
system  to  be  at  rest)  the  light  has  taken  exactly  the  same 
time  to  travel  from  C  to  A  as  to  travel  from  C  to  B. 

Now  let  us  suppose  that  an  observer  on  the  system  S  has 
watched  the  work  of  regulating  these  clocks  on  S'.  The  dis- 
tances CA  and  CB  appear  to  him  to  be 


instead  of  \d.  Moreover,  since  the  velocity  of  light  is  independ- 
ent of  the  velocity  of  the  source,  it  appears  to  him  that  the  light 
ray  proceeding  from  C  to  A  has  approached  A  at  the  velocity 
c+i>j  where  c  is  the  velocity  of  light,  while  the  ray  going  from 
C  to  £  has  approached  B  at  the  velocity  c—v.  Thus  to  him 
it  appears  that  the  light  has  taken  longer  to  go  from  C  to  B 
than  from  C  to  A  by  the  amount 


_ 
c  —  v  c-\-v  c2—v2 


THE   MEASUREMENT   OF   LENGTH   AND   TIME.  41 

But  since  $  =  v/c  the  last  expression  is  readily  found  to  be  equal 
to 

d 


Therefore,  to  an  observer  on  5  the  clocks  of  S'  appear  to  mark 
different  times;  and  the  difference  is  that  given  by  the  last 
expression  above. 

Thus  we  have  the  following  conclusion: 

THEOREM  VII.  Let  two  systems  of  reference  S  and  S'  have 
a  uniform  relative  -velocity  v.  Let  an  observer  on  S'  place  two  clocks 
at  a  distance  d  apart  in  the  line  of  relative  motion  of  S  and  Sf  and 
adjust  them  so  that  they  appear  to  him  to  mark  simultaneously 
the  same  hour.  Then  to  an  observer  on  S  the  clock  on  Sf  which 
is  forward  in  point  of  motion  appears  to  be  behind  in  point  of 
time  by  the  amount 

v         d 


where  c  is  the  velocity  of  light  and  $  =  v/c  (MVLR). 

It  should  be  emphasized  that  the  clocks  on  S'  are  in  agree- 
ment in  the  only  sense  in  which  they  can  be  in  agreement  for 
an  observer  on  that  system  who  supposes  (as  he  naturally  will) 
that  his  own  system  is  at  rest  —  notwithstanding  the  fact  that 
to  an  observer  on  the  other  system  there  appears  to  be  an 
irreconcilable  disagreement  depending  for  its  amount  directly 
on  the  distance  apart  of  the  two  clocks. 

According  to  the  result  of  the  last  theorem  the  notion  of 
simultaneity  of  events  happening  at  different  places  is  indefinite 
in  meaning  until  some  convention  is  adopted  as  to  how  simul- 
taneity is  to  be  determined.  In  other  words,  there  is  no  such 
thing  as  the  absolute  simultaneity  of  events  happening  at  different 
places. 

How  shall  we  adjust  this  remarkable  conclusion  to  our 
ordinary  intuitions  concerning  the  nature  of  time?  We  shall 
probably  most  readily  get  an  answer  to  this  question  by  inquir- 


42  THE   THEORY   OF   RELATIVITY. 

ing  further:  What  shall  we  mean  by  saying  that  two  events 
which  happen  at  different  places  are  simultaneous? 

First  of  all  it  should  be  noticed  that  we  have  no  direct  sense 
of  what  such  simultaneity  should  mean.  I  have  a  direct  per- 
ception of  the  simultaneity  of  two  events  in  my  own  con- 
sciousness. I  consider  them  simultaneous  because  they  are 
so  interlocked  that  I  cannot  separate  them  without  mutilating 
them.  If  two  things  happen  which  are  far  removed  from  each 
other  I  do  not  have  a  direct  perception  of  both  of  them  in  such 
way  that  I  perceive  them  as  simultaneous.  When  should  I 
consider  such  events  to  be  simultaneous? 

To  answer  this  question  we  are  forced  to  the  same  consider- 
ations as  those  wrhich  we  met  in  the  preceding  section.  There 
can  be  no  absolute  criterion  by  which  we  shall  be  able  to  fix 
upon  any  definition  as  the  only  appropriate  one.  We  must 
be  guided  by  the  demands  of  convenience,  and  by  this  alone. 

In  view  of  these  considerations  there  is  nothing  unthink- 
able about  the  conclusion  concerning  simultaneity  which  we 
have  obtained  above.  An  observer  A  on  one  system  of  refer- 
ence regulates  clocks  so  that  they  appear  to  him  to  be  simulta- 
neous. It  is  apparent  that  to  him  the  notion  of  simultaneity 
appears  to  be  entirely  independent  of  position  in  space.  His 
clocks,  even  though  they  are  separated  by  space,  appear  to 
him  to  be  running  together,  that  is,  to  be  together  in  a  sense 
which  is  entirely  independent  of  all  considerations  of  space. 

But  when  B  from  another  system  of  reference  observes 
the  clocks  of  A 's  system  they  do  not  appear  to  him  to  be  mark- 
ing simultaneously  the  same  hour;  and  their  lack  of  agreement 
is  proportional  to  their  distance  apart,  the  factor  of  propor- 
tionality being  a  function  of  the  relative  velocity  of  the  two 
systems. 

Thus  instants  of  time  at  different  places  which  appear  to 
A  to  be  simultaneous  in  a  sense  which  is  entirely  independent 
of  all  considerations  of  space  appear  to  B  in  a  very  different 
light;  namely,  as  if  they  were  different  instants  of  time,  the 
one  preceding  the  other  by  an  amount  directly  proportional 


THE  MEASUREMENT   OF  LENGTH  AND  TIME.  43 

to  the  distance  between  the  points  in  space  at  which  events 
occur  to  mark  these  instants.  Even  the  order  of  succession 
of  events  is  in  certain  cases  different  for  the  two  observers,  as 
one  can  readily  verify. 

It  thus  appears  that  the  notion  of  simultaneity  at  different 
places  is  relative  to  the  system  on  which  it  is  determined.  The 
only  meaning  which  it  can  have  is  that  which  is  given  to  it  by 
convention. 


CHAPTER  IV. 
EQUATIONS  OF  TRANSFORMATION. 

§20.  TRANSFORMATION  OF  SPACE  AND  TIME  COORDINATES. 

IT  is  now  an  easy  matter  to  derive  the  Einstein  formulae 
for  the  transformation  of  space  and  time  coordinates.  Let 
two  systems  of  reference  S  and  S'  have  the  relative  velocity 
v  in  the  line  /.  Let  systems  of  rectangular  coordinates  be 
attached  to  the  systems  of  reference  S  and  S'  in  such  a  way 
that  the  #-axis  of  each  system  is  in  the  line  /,  and  let  the  ^-axis 
and  the  s-axis  of  one  system  be  parallel  to  the  ^-axis  and  the 
s-axis  respectively  of  the  other  system.  Let  the  origins  of  the 
two  systems  coincide  at  the  time  t  =  o.  Furthermore,  for  the 
sake  of  distinction,  denote  the  coordinates  on  6*  by  x,  y,  z,  t 
and  those  on  S'  by  x' ',  y' ',  2',  t'.  We  require  to  find  the  value 
of  the  latter  coordinates  in  terms  of  the  former. 

From  postulate  L  it  follows  at  once  that  y'  —y  and  2' =2. 
Let  an  observer  on  5  consider  a  point  which  at  time  t  =  o  appears 
to  him  to  be  at  distance  *  x  from  the  ;yY-plane;  at  time  t  —  t 
it  will  appear  to  him  to  be  at  the  distance  x—vt  from  the  y'z'- 
plane.  Now,  by  an  observer  on  S'  this  distance  is  denoted 
by  x'.  Then  from  theorem  VI  we  have 


Now  consider  a  point  at  the  distance  x  from  the  ys-plane  at 
time  t  =  t  in  units  of  system  S.     From   theorem  VII  it  follows 

*  The  algebraic  sign  of  the  distance  is  supposed  to  be  taken  into  account  in 
the  value  of  x. 

44 


EQUATIONS   OF  TRANSFORMATION.  45 

that  to  an  observer  on  S  the  clock  on  S'  at  the  same  distance 
x  from  the  yz-plane  will  appear  behind  by  the  amount 


where  c  is  the  velocity  of  light.    That  is  to  say,  in  units  of  5 
this  clock  would  register  the  time 

v_ 
t--2x. 

Hence,  by  means  of  theorem  IV,  we  have  at  once  the  result 

/  / ^ 

c2 

Solving  the  two  equations  involving  x'  and  tf  and  collecting 
results,  we  have 

''=v=rpH*)' 

(A)  x'  =  -J=.(x-vt),  (MVLR) 


where  $  =  v/c  and  c  is  the  velocity  of  light. 

In  the  same  way  we  may  obtain  the  equations  which  express 
/,  x,  y,  z  in  terms  of  t' ,  x' ,  y',  zf.  But  these  can  be  found  more 
easily  by  solving  equations  (.4)  for  t,  x,  y,  z.  Thus  we  have 


(Ai)  x=-j==(x'+vt'),  (MVLR) 

y=y', 

These  two  sets  of  equations  (A)  and  (Ai)  are  identical  in  form 
except  for  the  sign  of  v.  This  symmetry  in  the  transforma- 
tions constitutes  one  of  their  chief  points  of  interest. 


46  THE  THEORY  OF  RELATIVITY. 

§21.  THE  ADDITION  OF  VELOCITIES. 

We  shall  now  derive  the  formulae  for  the  addition  of  veloc- 
ities. 

Let  the  velocity  of  a  point  in  motion  be  represented  in  units 
belonging  to  S'  and  to  5  by  means  of  the  equations 

x'  =  ux't',     y'  =  uvt',      z'  =  uz't'\ 
x  =  uxt,         y  =  uyt,         z  =  uzt, 

respectively.  In  the  first  of  these  substitute  for  /',  x',  y' ',  z' 
their  values  given  by  (^4),  solve  for  x/t,  y/t,  z/t  and  replace 
these  quantities  by  their  equals  ux,  uy,  uz  respectively.  Thus 
we  have 


ux  = 


VUX'' 


(B)  u,  =  ue',  (MVLR) 


From  these  results  it  follows  that  the  law  of  the  parallelo- 
gram of  velocities  is  only  approximate.  This  conclusion  of 
the  theory  of  relativity  has  given  rise,  in  the  minds  of  some 
persons,  to  the  most  serious  objections  to  the  entire  theory. 

Suppose  that  both  the  velocities  considered  above  are  in 
the  line  of  relative  motion  of  61  and  5".  Then  we  have 

v+u' 


This  equation  gives  rise  to  the  following  theorem: 

THEOREM  VIII.    If  two  velocities,  each  of  which  is  less  than 
c,  are  combined  the  resultant  "velocity  is  also  less  than  c(MVLR). 


EQUATIONS   OF   TRANSFORMATION.  47 

To  prove  this  we  substitute  in  the  preceding  equation  for 
v  and  u'  the  values 

v  =  c  —  k,     ur  =  c  —  l 

where  each  of  the  numbers  k  and  /  is  positive  and  less  than  c. 
Then  the  equation  becomes 

2c  —  k  —  I 


u  —  c 


2C-k-l+~ 
C 


The  second  member  is  evidently  less  than  c.    Hence  the  theorem. 

If,  however,  either  one  (or  both)  of  the  velocities  v  and  ur 
is  equal  to  c — and  hence  k  or  /  (or  both)  is  equal  to  zero — we 
see  at  once  from  the  last  equation  that  u  =  c.  Hence,  we  have 
the  following  result: 

THEOREM  IX.  If  a  velocity  c  is  compounded  with  a  velocity 
equal  to  or  less  than  c,  the  resultant  velocity  is  c(MVLR). 

§22.  MAXIMUM  VELOCITY  OF  A  MATERIAL  SYSTEM. 

A  conclusion  of  importance  is  implicity  involved  in  the  pre- 
ceding results.  It  can  probably  be  seen  in  the  simplest  way 
by  reference  to  the  first  two  equations  (A),  these  being  nothing 
more  nor  less  than  an  analytic  formulation  of  theorems  IV 
and  VI.  If  P  is  in  numerical  value  greater  than  i — whence 
i  —  g2  is  negative — the  transformation  of  time  coordinates 
from  one  system  to  the  other  gives  an  imaginary  result  for 
the  time  in  one  system  if  the  time  in  the  other  system  is  real. 
Likewise,  measurement  of  length  in  the  direction  of  motion 
is  imaginary  in  one  system  if  it  is  real  in  the  other.  Both  of 
these  conclusions  are  absurd  and  hence  the  numerical  value  of 
g  is  equal  to  or  less  than  i.  If  it  is  i,  then  any  length  in  one 
system,  however  short,  would  be  measured  in  the  other  as  infin- 
ite; and  a  like  result  holds  for  time.  Hence  g  is  numerically 
less  than  i.  But  $  =  v/c,  the  ratio  of  the  relative  velocity  of 
the  two  systems  to  the  velocity  of  light.  Hence: 


48  THE   THEORY   OF   RELATIVITY. 

THEOREM  X.  The  velocity  of  light  is  a  maximum  which 
the  velocity  of  a  material  system  may  approach  but  can  never 
reach  (MVLR). 

It  should  be  pointed  out  that  this  theorem  may  also  be  proved 
directly  by  means  of  theorem  IX. 


§  23.    TIME  AS  A  FOURTH  DIMENSION. 

I  have  no  intention  of  asserting  that  time  is  a  fourth  dimen- 
sion of  space  in  the  sense  in  which  we  ordinarily  employ  the 
word  "  dimension";  such  a  statement  would  have  no  meaning. 
I  wish  to  point  out  rather  that  it  is  in  some  measure  connected 
with  space,  and  that  in  many  formulae  it  must  enter  as  it  would 
if  it  were  essentially  and  only  a  fourth  dimension. 

We  shall  see  this  readily  if  we  examine  the  formulae  (A) 
of  transformation  from  one  system  of  reference  to  another. 
Here  the  time  variable  /  enters  in  a  way  precisely  analogous  to 
that  in  which  the  space  variables  #,  y,  z  enter. 

Suppose  now  that  the  law  of  some  phenomenon  as  observed 
on  S'  is  given  by  the  equation 

F(*',/,s',O=o 

and  we  desire  to  know  the  expression  of  this  law  on  S.  We 
substitute  for  x' ',  y',  z',  t'  their  values  in  terms  of  x,  y,  z,  t 
given  in  (A);  and  thus  we  obtain  an  equation  stating  the  law 
in  question. 

From  these  considerations  it  appears  that  in  many  of  our 
problems,  namely  in  those  which  have  to  do  at  once  with  two 
or  more  systems  of  reference,  the  time  and  space  variables 
taken  together  play  the  role  of  four  variables  each  having  to 
do  with  one  dimension  of  a  four-dimensional  continuum. 

This  conclusion  raises  philosophical  questions  of  profound 
importance  concerning  the  nature  of  space  and  time;  but  into 
these  we  cannot  enter  here. 


CHAPTER  V. 
MASS  AND  ENERGY. 

§  24.    DEPENDENCE  OF  MASS  ON  VELOCITY. 

SUPPOSE  that  we  have  two  systems  of  reference  Si  and  S2 
moving  with  a  relative  velocity  v.  We  inquire  as  to  whether, 
and  in  what  way,  the  mass  of  a  body  as  measured  on  the  two 
systems  depends  on  v.  Will  a  given  body  have  the  same  measure 
of  mass  when  that  mass  is  estimated  in  units  of  Si  and  in 
units  of  6*2?  And  will  the  mass  of  a  body  depend  on  the  direc- 
tion of  its  motion  by  means  of  which  that  mass  is  measured? 
Our  purpose  in  this  section  is  to  answer  these  two  questions. 

The  two  most  important  directions  in  which  to  measure 
the  mass  of  a  body  are,  first,  that  perpendicular  to  the  line  of 
relative  motion  of  Si  and  £2,  and,  secondly,  that  parallel  to 
this  line  of  motion.  For  convenience  in  distinguishing  these 
we  shall  speak  of  the  "  transverse  mass  "  of  a  body  as  that  with 
which  we  have  to  deal  when  we  are  concerned  with  the  motion 
of  the  body  in  a  direction  perpendicular  to  the  line  of  relative 
motion  of  Si  and  52 ;  when  the  motion  is  parallel  to  this  line 
we  shall  speak  of  the  "  longitudinal  mass  "  of  the  body. 

Lewis  and  Tolman  (Phil.  Mag.  18:  510-523)  determine 
what  they  call  the  "  mass  of  a  body  in  motion,"  employing 
for  this  purpose  a  very  simple  and  elegant  method.  This 
"  mass  "  is  what  we  have  just  defined  as  the  transverse  mass 
of  the  body.  We  employ  the  excellent  method  of  these  authors 
in  deriving  the  formula  for  transverse  mass. 

Suppose  that  an  experimenter  A  on  the  system  Si  constructs 
a  ball  Bi  of  some  rigid  elastic  material,  with  unit  volume,  and 

49 


50  THE   THEORY  OF   RELATIVITY. 

puts  it  in  motion  with  unit  velocity  in  a  direction  perpendicular 
to  the  line  of  relative  motion  of  Si  and  62,  the  units  of  measure- 
ment employed  being  those  belonging  to  Si.  Likewise  sup- 
pose that  an  experimenter  C  on  £2  constructs  a  ball  B2  of  the 
same  material,  also  of  unit  volume,  and  puts  it  in  motion  with 
unit  velocity  in  a  direction  perpendicular  to  the  line  of  relative 
motion  of  Si  and  6*2;  we  suppose  that  the  measurements  made 
by  C  are  with  units  belonging  to  52.  Assume  that  the  exper- 
iment has  been  so  planned  that  the  balls  will  collide  and  rebound 
over  their  original  paths,  the  path  of  each  ball  being  thought 
of  as  relative  to  the  system  to  which  it  belongs. 

Now  the  relation  of  the  ball  B2  to  the  system  Si  is  the  same 
as  that  of  the  ball  BI  to  the  system  6*2,  on  account  of  the  perfect 
symmetry  which  exists  between  the  two  systems  of  reference 
in  accordance  with  previous  results.  Therefore  the  change  of 
velocity  of  B2  relative  to  its  starting  point  on  6*2  as  measured 
by  A  is  equal  to  the  change  of  velocity  of  BI  relative  to  its 
starting  point  on  Si  as  measured  by  C.  Now  velocity  is  equal 
to  the  ratio  of  distance  to  time :  and  in  the  direction  perpendicular 
to  the  line  of  relative  motion  of  the  two  systems  the  units  of 
length  are  equal;  but  the  units  of  time  are  unequal.  Hence 
to  either  of  the  observers  the  change  of  velocity  of  the  two 
balls,  each  with  respect  to  its  starting  point  on  its  own  system, 
will  appear  to  be  unequal. 

To  A  the  time  unit  on  62  appears  to  be  longer  than  his  own 
in  the  ratio  i  :  Vi  —  g2  (see  theorem  IV).  Hence  to  A  it 
must  appear  that  the  change  in  velocity  of  #2  relative  to  its 
starting  point  is  smaller  than  that  of  BI  relative  to  its  starting 
point  in  the  ratio  vY—  $2  :  i.  But  the  change  in  velocity  of 
each  ball  multiplied  by  its  mass  gives  its  change  in  momentum. 
From  postulate  Ci  it  follows  that  these  two  changes  of  momentum 
are  equal.  Hence  to  A  it  appears  that  the  mass  of  the  ball 
BI  is  smaller  than  that  of  the  ball  B2  in  the  ratio  Vi  —  $2  :  i. 

Similarly,  it  may  be  shown  that  to  C  it  appears  that  the 
mass  of  the  ball  B2  is  smaller  than  that  of  BI  in  the  ratio 

VrT2  :  i. 


MASS   AND   ENERGY.  51 

From  our  general  results  concerning  the  measurement  of 
length  it  follows  that  if  the  ball  which  has  been  constructed 
by  A  were  transferred  to  C's  system  it  would  be  impossible 
for  C  to  distinguish  -4's  ball  from  his  own  by  any  considerations 
of  shape  and  size.  Likewise,  as  A  looks  at  them  from  his  own 
system  he  is  similarly  unable  to  distinguish  them.  It  is  there- 
fore natural  to  take  the  mass  of  C's  ball  as  that  which  ^4's 
would  have  if  it  had  the  velocity  v  with  respect  to  Si  of  the 
system  52.  Thus  we  obtain  a  relation  existing  between  the 
mass  of  a  body  in  motion  and  at  rest. 

Now,  "  mass  "  as  we  have  measured  it  above  is  the  trans- 
verse mass  of  our  definition.  From  the  argument  just  carried 
out  we  are  forced  to  conclude  that  the  transverse  mass  of  a  body 
in  motion  depends  (in  a  certain  definite  way)  on  the  velocity 
of  that  motion.  The  result  may  be  formulated  as  follows: 

THEOREM  XI.  Let  mo  denote  the  mass  of  a  body  when  at  rest 
relative  to  a  system  of  reference  S.  When  it  is  moving  with  a 
velocity  v  relative  to  S  denote  by  t(mv)  its  transverse  mass,  that  is, 
its  mass  in  a  direction  perpendicular  to  its  line  of  motion.  Then 
we  have 


where  ^  —  D[C  and  c  is  the  velocity  of  light  (MVLRCi). 

In  the  statement  of  this  theorem  we  have  tacitly  assumed 
that  the  mass  of  a  body  at  rest  relative  to  S,  when  measured 
by  means  of  units  belonging  to  S,  is  independent  of  the  direc- 
tion in  which  it  is  measured.  If  this  assumption  were  not 
true  we  should  have  a  means  of  detecting  the  motion  of  5, 
a  conclusion  which  is  in  contradiction  to  postulate  M. 

In  order  to  find  the  longitudinal  mass  of  a  moving  body 
we  first  find  the  relation  which  exists  between  longitudinal 
mass  and  transverse  mass.  We  employ  for  this  purpose  the 
elegant  method  of  Bumstead  (Am.  Journ.  Science  (4)  26: 
498-500). 

Let  us  as  usual  consider  two  systems  of  reference  Si  and 
52  moving  with  a  relative  velocity  v,  observers  A  and  B  being 


52  THE  THEORY  OF  RELATIVITY. 

stationed  on  Si  and  £2  respectively.  Suppose  that  B  per- 
forms the  following  experiment:  He  takes  a  rod  of  two  units 
length,  whose  mass  is  so  small  as  to  be  negligible,  and  attaches 
to  its  ends  two  balls  of  equal  mass.  Then  he  suspends  this 
rod  by  a  wire  so  as  to  form  a  torsion  pendulum.  We  assume 
that  the  line  of  relative  motion  of  the  two  systems  is  perpendic- 
ular to  the  line  of  this  wire. 

Let  us  consider  the  period  of  this  torsion  pendulum  in  the 
two  cases  when  the  rod  is  clamped  to  the  wire  so  as  to  be  in 
equilibrium  in  each  of  the  following  two  positions:  (i)  With 
its  length  perpendicular  to  the  line  of  relative  motion  of  Si 
and  62;  (2)  with  its  length  parallel  to  this  line  of  motion. 

As  B  observes  it  the  period  must  be  the  same  in  the  two 
cases;  for,  otherwise,  he  would  have  a  means  of  detecting  his 
motion  by  observations  made  on  his  system  alone,  contrary  to 
postulate  M.  Then  from  the  relation  of  time  units  on  Si 
and  52  it  follows  that  the  two  periods  will  also  appear  the  same 
to  A.  As  observed  by  B  the  apparent  mass  of  the  balls  is 
the  same  in  both  cases.  We  inquire  as  to  how  they  appear 
to  A.  Let  mi  and  mz  be  the  apparent  masses,  as  observed  by 
A,  in  the  first  and  second  cases  respectively.  It  is  obvious 
that  mi  is  the  longitudinal  mass  and  m^  the  transverse  mass 
of  the  balls  in  question. 

When  the  pendulum  is  in  motion  it  appears  to  B  that  each 
ball  traces  a  circular  arc.  From  the  relations  between  the  units 
of  length  in  the  two  systems  it  follows  that  to  A  it  appears 
that  the  balls  trace  arcs  of  an  ellipse  whose  semiaxes  are  i 
and  Vi  —  £2  and  lie  perpendicular  and  parallel,  respectively, 
to  the  line  of  relative  motion  of  the  two  systems. 

Let  us  now  determine  the  period  of  each  of  these  two 
pendulums  as  they  are  observed  by  A.  By  equating  the  expres- 
sions for  these  periods  we  shall  find  the  relation  which  exists 
between  m\  and  m^. 

Let  x  and  y  be  the  cartesian  coordinates  of  a  point  as  deter- 
mined by  A.  the  axes  of  reference  being  the  major  and  minor 
axes  of  the  ellipse  in  which  the  balls  move.  Let  x'  and  y'  be 


MASS  AND  ENERGY.  53 

the  coordinates  of  the  same  point  as  determined  by  B.    Then 
the  circular  path  of  motion,  as  determined  by  B,  has  the  equations 

#'  =  cos  0,  y=sin  0, 

the  angle  6  being  measured  from  the  major  axis  of  the  ellipse. 
The  equations  of  the  ellipse,  as  determined  by  A ,  are 


#  =  cos  0,  y  =     i  —  g2  sin  0. 

In  the  first  case — when  the  rod  is  perpendicular  to  the 
line  of  relative  motion  of  Si  and  £2 — the  amount  of  twisting 
in  the  wire  when  the  ball  is  in  a  given  position  is  the  numerical 
value  of  the  corresponding  angle  0;  and  therefore  the  potential 
energy  *  is  proportional  to  02,  say  that  it  is  |&02.  Now  from 
the  values  of  y  and  x  above  we  have 


y=xVi  —  g2  tan  0. 
For  small  oscillations  we  have  x  =  i  and  tan  0  =  0;  and  therefore 


Hence  the  potential  energy  is 

,     k     2> 

and  the  equation  of  motion  of  the  particle  becomes 

d?y_         k 

Hence  the  period  T\  of  oscillation  is 
Ti  =  : 


*That  the  potential  energy  is  proportional  to  62  when  measured  byB  is 
obvious.  Since  A  observes  a  different  apparent  angle  6'  (say)  corresponding  to 
B's  observed  angle  0  it  might  at  first  sight  appear  that  the  potential  energy  as 
observed  by  A  is  proportional  to  6' 2;  that  this  is  not  the  case  is  seen  from  the 
fact  that  for  a  given  twist  in  the  wire  6'  depends  on  the  direction  of  equilibrium 
of  the  bar,  that  is,  it  depends  on  the  way  in  which  the  bar  is  attached  to  the  wire; 
hence,  if  the  potential  energy  as  observed  by  A  were  proportional  to  6'2,  it 
would  depend  on  the  way  in  which  the  bar  is  attached.  Since  this  is  obviously 
not  the  case  we  conclude  that  the  potential  energy  is  proportional  to  02. 


54  THE   THEORY  OF  RELATIVITY. 

In  the  second  case — when  the  rod  is  parallel  to  the  line  of 
relative  motion  of  Si  and  £2 — the  amount  of  twisting  in  the 
wire  for  a  given  position  of  the  balls  is  the  numerical  value 

(\  2 
—  -ej  .    We  have 

x=    ,— — cot  0. 

Vl-g2 

For  small  oscillations  we  have 


TU 

cot  0  =  tan  —  —  0    =  —  — 


Hence   the  potential   energy  is  %kx2,   and   the  period   Tz   of 
oscillation  is  therefore 


Equating  the  two  periods  of  oscillation  found  above  we  have 


Remembering  that  mi  and  m^  are  the  longitudinal  mass  and 
the  transverse  mass,  respectively,  and  making  use  of  theorem 
XI,  we  have  the  following  result: 

THEOREM  XII.  Let  mo  denote  the  mass  of  a  body  when  at 
rest  relative  to  a  system  of  reference  S.  When  it  is  moving  with 
a  velocity  v  relative  to  S  denote  by  l(mv)  its  longitudinal  mass, 
that  is,  its  mass  in  a  direction  parallel  to  its  line  of  motion.  Then 
we  have 

K    \         m° 

/W  =  (I_g2)f 

where  $=v/c  and  c  is  the  velocity  of  light  (MVLRCiCz). 


§25.    ON  THE  DIMENSIONS  OF  UNITS. 

Denote  the  fundamental  measurable  physical  entities  mass, 
length  and  time  by  M ,  L  and  T  respectively.  Then  the  definition 
of  derived  entities  gives  rise  to  the  so-called  dimensional  equa- 


MASS   AND   ENERGY.  55 

tions.  Thus  if  V  denote  velocity  we  have  the  dimensional 
equation 

F  =  - 

r 

That  such  equations  must  be  useful  in  obtaining  the  relations 
of  units  in  two  systems  of  reference  is  obvious.  Thus  from 
the  above  dimensional  equation  for  V  we  may  at  once  derive 
the  fundamental  result  (see  theorem  VI)  concerning  the  relation 
of  units  of  length  in  the  line  of  relative  motion  of  two  systems 
not  at  rest  relatively  to  each  other.  For  this  purpose  it  is 
sufficient  to  employ  postulate  V  and  theorem  IV.  The  reader 
can  easily  supply  the  argument.  Or,  conversely,  if  one  knows 
the  relations  which  exist  between  units  of  length  and  units 
of  time  in  two  systems  one  concludes  readily  to  the  truth  of 
postulate  V. 

Likewise,  from  the  dimensional  equation 

acceleration  =  ™, 

one  may  readily  determine  the  relations  which  exist  between 
units  of  acceleration  on  two  systems,  it  being  assumed  that 
the  relations  of  time  units  and  length  units  are  known.  Making 
this  assumption,  then,  the  two  dimensional  equations  above 
give  us  the  following  theorem : 

THEOREM  XIII.  Let  two  systems  Si  and  82  move  with  a 
relative  velocity  v  in  the  direction  of  a  line  /,  and  let  $  =  v/c  where 
c  is  the  velocity  of  light.  Then  to  an  observer  on  Si  it  appears  that 
the  unit  of  velocity  [acceleration}  on  Si  bears  to  the  unit  of  velocity 
[acceleration]  on  82  the  ratio  i  :  i[i  :  Vi  —  $2]  or  i  :  Vi  —  g2 
[i  :  i  —  £2]  according  as  the  motion  is  parallel  to  I  or  perpendicular 
to  l(MVLR). 

Let  us  use  F  to  denote  force.  Then  from  the  dimensional 
equation 

77      ML        ' 
b~-~--^2> 

we  shall  be  able  to  draw  an  interesting  conclusion  concerning 
the  measurement  of  force. 


56  THE   THEORY   OF   RELATIVITY. 

Suppose  that  an  observer  B  on  a  system  £2  carries  out 
some  observations  concerning  a  certain  rectilinear  motion, 
measuring  the  quantities  Mf,  Z/,  Tf,  so  that  he  has  the  equation 

M>L> 


Another  observer  A  on  a  system  S\  (having  with  respect  to  52 
the  velocity  v  in  the  line  /)  measures  the  same  force  calling 
it  F.  Required  the  value  of  F  in  terms  of  F',  when  the  motion 
is  parallel  to  /  and  when  it  is  perpendicular  to  /,  the  estimate 
being  made  by  A  . 

When  the  motion  is  perpendicular  to  /  —  that  is,  when  the 
force  acts  in  a  line  perpendicular  to  /  —  we  have 


ML    M'Vi-2-!/         F' 


When  the  motion  is  parallel  to  /  we  have 

_      ML    M'(i 


These  results  may  be  stated  in  the  following  theorem: 

THEOREM  XIV.  In  the  same  systems  of  reference  as  in  theorem 
XIII,  let  an  observer  on  6*2,  measure  a  given  force  F'  in  a  direction 
perpendicular  to  I  and  in  a  direction  parallel  to  /,  and  let  F\  andFz 
be  the  values  of  this  force  as  measured  in  the  first  and  second  cases 
respectively  by  an  observer  on  Si.  Then  we  have 

Fi=    Z— ,    F2  =  (i-^)Ff     (MVLRdC2). 
vi  —  @2 

It  is  obvious  that  a  similar  use  may  be  made  of  the  dimen- 
sional equation  of  any  derived  unit  in  determining  the  relation 
which  exists  between  this  unit  in  two  relatively  moving  systems 
of  reference. 


MASS  AND  ENERGY.  57 

§26.  MASS  AND  ENERGY. 

If,  as  is  frequently  done,  we  employ  for   the  definition  of 
the  kinetic  energy  E  the  relation  (compare  §  14) 


E  =  ^Mdi)  =  \  mvdv, 


it  is  clear  that  for  the  mass  m  we  should  take  the  longitudinal 
mass  l(mv).  Then  let  mo  denote  the  mass  of  the  body  at  rest, 
EQ  its  energy  when  at  rest  (that  is,  the  energy  due  to  its  internal 
activity),  and  Ev  its  energy  when  moving  at  the  velocity  v. 
Then  clearly  E=Ev—Eo,  so  that  in  view  of  theorem  XII  we 
have 

E=EC-EO=JO  (!_p2)|J 

whence,  on  integration,  we  have 


Hence  for  the  kinetic  energy  of  a  moving  body  we  have 

E  =  woc2(i&2+i&4+  •  •  .  ); 
or,  to  a  first  approximation  only, 

-E  =  ^7Wo^2. 

Therefore  the  usual  formula  for  kinetic  energy  in  the  classical 
mechanics  is  only  a  first  approximation. 

Since  relation  (i)  is  to  be  true  for  all  values  of  v  it  is  obvious 
that  we  have 

Ec=    ,  -  =+k,    Eo  =  m0c2+k, 

vi  —  p2 

where  k  is  a  constant,  that  is,  a  quantity  independent  of  v. 
From  the  first  of  these  equations  we  conclude  further  that 


so  that  the  total  energy  of  a  body,  decreased  by  the  constant  k, 
is  directly  proportional  to  its  transverse  mass.  In  case  the  body 
is  at  rest  its  mass  in  one  direction  is  the  same  as  in  another; 


58  THE   THEORY   OF   RELATIVITY. 


hence  wo  =  /(WQ).  Bearing  this  in  mind  we  have  the  following 
theorem: 

THEOREM  XV.  Let  mo  be  the  mass  of  a  body  when  at  rest 
with  respect  to  a  given  system  of  reference  and  let  t(mv)  denote 
its  transverse  mass  when  it  is  moving  with  the  velocity  v  (the  case 
v  =  o  is  not  excluded)  .  Then  the  total  energy  Ev  which  it  possesses 
is  c2't(mc)+k,  where  k  is  a  constant. 

§27.    ON  MEASURING  THE  VELOCITY  OF  LIGHT. 

The  following  relations  are  immediate  consequences  of 
equations  written  out  above: 

Ev  —  Ep  Ep  —  k 

—  =c2,     Ev  —  k=, 
t(mv)-m0  Vi-$2 

Now,  suppose  that  an  experimenter  contributes  to  a  body 
which  is  at  rest  a  known  amount  of  energy  and  determines 
the  velocity  which  this  causes  the  body  to  acquire.  If  the 
two  measurements  are  made  with  sufficient  accuracy  one  will 
be  able,  by  substituting  the  results  in  the  first  of  the  above 
equations,  to  determine  in  this  way  the  velocity  of  light. 
Actually  to  carry  out  this  remarkable  method  for  measuring 
c  would  doubtless  be  very  difficult;  but  the  obvious  great 
importance  of  the  result  is  certainly  such  as  to  justify  a  care- 
ful consideration  of  the  problem.  If  the  value  of  c  determined 
in  this  way  should  agree  well  with  its  value  as  otherwise  found, 
this  would  give  us  an  interesting  confirmation  of  the  theory 
of  relativity. 

Let  us  consider  the  mass  of  a  rotating  top,  the  mass  being 
measured  along  the  axis  of  rotation.  According  to  our  results 
this  mass  should  be  different  from  that  of  the  same  top  when 
at  rest,  and  the  difference  should  bear  a  definite  relation  to 
the  amount  of  energy  which  is  involved  in  the  rotation.  If 
the  measurements  here  involved  could  be  made  with  sufficient 
accuracy  we  would  have  another  means,  independent  of  light 
itself,  for  the  measurement  of  the  light-velocity  c.  Again, 
this  experiment  would  afford  us  a  measure  of  transverse  mass 


MASS   AND   ENERGY.  59 

and  in  that  way  could  lead  to  a  confirmation  of  the  theory  of 
relativity,  provided  that  we  assume  c  as  known  from  inde- 
pendent measurements;  and  this  confirmation,  it  is  to  be 
noticed,  would  be  independent  of  electrical  considerations. 

Remark. — It  seems  to  be  impossible  to  determine  the  con- 
stant k  which  enters  into  the  above  discussion.  But  in  the 
absence  of  any  evidence  to  the  contrary  it  would  appear  natural 
tentatively  to  assume  that  k  is  zero.  On  the  basis  of  this 
assumption  we  should  have  the  following  remarkable  con- 
clusions: The  mass  of  a  body  at  rest  is  simply  the  measure 
of  its  internal  energy.  The  transverse  mass  of  a  body  in  motion 
is  the  measure  of  its  internal  energy  and  its  kinetic  energy 
taken  together.  Its  longitudinal  mass  is  its  total  energy  mul- 
tiplied by  a  simple  factor.  One  can  hardly  resist  the  conclusion 
that  the  transverse  mass  of  a  body  depends  entirely  on  its 
energy,  and  therefore  that  matter  is  merely  one  manifestation 
of  energy. 

§28.    ON  THE  PRINCIPLE  OF  LEAST  ACTION. 

In  §  26  we  saw  that  in  the  theory  of  relativity  the  classical 
formula  E  =  %mv2  for  the  measure  of  kinetic  energy  is  true 
only  as  a  first  approximation.  This  is  due  to  the  fact  that 
mass  is  a  variable  quantity.  But  the  conclusion  does  not 
appear  to  necessitate  our  surrender  of  the  law  of  conservation 
of  energy. 

The  same  causes  which  lead  to  a  modification  in  the  formula 
for  E  will  also  require  a  corresponding  modification  in  the 
value  of  the  action  A  as  defined  in  §  14.  The  question  arises 
as  to  whether  the  principle  of  least  action  is  left  intact.  I 
cannot  enter  upon  the  investigation  here;  but  the  problem 
seems  to  me  to  be  of  importance  and  consequently  I  am  stating 
it  in  the  hope  that  some  one  will  be  led  to  consider  the  solution. 

Undoubtedly  the  principle  of  least  action  is  one  which  should 
be  given  up  only  when  there  are  strong  reasons  for  it.  It  is 
a  mathematical  formulation  of  the  law  that  nature  accom- 
plishes her  ends  with  the  least  expenditure  of  labor,  so  to 


60  THE   THEORY  OF  RELATIVITY. 

speak.  Certainly  this  law  is  one  which  appeals  to  our  minds 
with  strong  force.  There  is  something  about  it  which  is  aesthet- 
ically satisfying  in  a  high  degree.  It  seems  to  me,  however, 
that  a  fresh  study  of  it  should  be  made  in  the  light  of  the 
theory  of  relativity. 

§29.    A  MAXIMUM  VELOCITY  FOR  MATERIAL  BODIES. 

There  are  several  ways  by  which  it  may  be  shown  that 
a  material  body  cannot  have  a  velocity  as  great  as  that  of 
light.  One  of  these  we  used  in  §  22,  showing  that,  if  a  material 
body  had  a  velocity  greater  than  that  of  light,  the  numerical 
measure  of  length  and  time  on  that  body  would  be  imaginary, 
while  if  its  velocity  were  just  equal  to  that  of  light  a  given 
time  interval  would  have  an  infinite  measure. 

We  may  also  prove  the  same  theorem  by  means  of  a  con- 
sideration of  mass.  Let  us  consider  the  equation 


where  mo  is  the  mass  of  a  body  at  rest  relative  to  a  given 
system  of  reference  S  and  l(mv)  is  the  longitudinal  mass  of 
the  body  moving  with  a  velocity  v  with  respect  to  S.  If  we 
consider  larger  and  larger  values  of  the  velocity  v  we  see  that 
l(mv)  increases  and  becomes  infinite  as  v  approaches  c.  This 
is  equivalent  to  saying  that  the  longitudinal  mass  of  any  material 
body  becomes  infinite  as  the  velocity  of  that  body  approaches 
c.  Therefore  it  would  require  an  infinite  force  to  give  to  a 
material  body  the  velocity  c\  that  is,  c  is  a  maximum  velocity 
which  the  velocity  of  a  material  body  may  approach  but  can 
never  reach. 

§30.    ON  THE  NATURE  OF  MASS. 

This  conclusion  concerning  the  maximum  velocity  of  a 
material  body  brings  up  important  considerations  concerning 
the  essential  nature  of  mass  and  material  things.  How  shall 
we  conceive  of  matter  so  that  it  should  have  this  astonishing 
property? 


MASS   AND   ENERGY.  61 

In  the  present  state  of  science  any  answer  to  this  question 
must  necessarily  be  of  a  speculative  character;  but  it  is  probably 
worth  while  to  mention  briefly  a  theory  of  mass  which  is  con- 
sistent with  the  existence  of  a  maximum  velocity  for  a  material 
body. 

Let  us  suppose  that  the  mass  of  a  piece  of  matter  is  due 
to  a  kind  of  strain  in  the  ether,  and  that  this  strain  is  prin- 
cipally localized  in  a  relatively  small  portion  of  space,  but  that 
from  this  center  of  localization  there  go  out  to  infinity  in  all 
directions  lines  of  strain  which  belong  essentially  to  the  piece 
of  matter.  (We  make  no  assumption  as  to  how  this  strain 
is  set  up;  it  may  be  due  largely  or  entirely  to  the  motion  of 
electrons  in  the  molecules  of  the  matter.)  Suppose  that  these 
lines  of  strain,  except  in  the  immediate  neighborhood  of  the 
center  of  localization,  are  of  such  nature  as  to  escape  detection 
by  our  usual  methods.  Suppose  further  that  when  the  piece  of 
matter  is  moved,  that  is,  when  the  center  of  localization  is 
displaced,  these  lines  of  strain  have  a  corresponding  displace- 
ment, but  that  the  ether  of  space  resists  this  displacement, 
the  degree  of  resistance  depending  on  the  velocity. 

If  the  mass  of  matter  is  due  to  such  a  strain  in  the  ether 
it  is  natural  to  suppose  that  mass  is  a  measure  of  the  amount 
of  that  strain.  But,  on  our  present  hypothesis,  we  see  that 
when  matter  is  moved  through  space  there  is  an  increase  of  the 
strain  on  the  ether  due  to  such  motion.  This  manifests  itself 
to  us  in  the  way  of  an  increase  in  the  mass  of  the  given  piece 
of  matter. 

Moreover,  when  the  body  is  in  motion  it  is  natural  to  suppose 
that  these  lines  of  strain  are  not  distributed  evenly  in  all  direc- 
tions. On  account  of  this  fact  it  would  not  be  a  matter  for 
surprise  if  the  mass  of  a  moving  body  were  different  in  different 
directions. 

It  thus  appears  that  appropriate  hypotheses  (which  have 
nothing  in  them  inherently  unnatural)  would  lead  us  to  expect 
the  same  descriptive  properties  of  mass  as  those  which  are 
actually  found  to  exist  if  one  accepts  the  postulates  of  rela- 
tivity. Hence  we  conclude  that  there  is  nothing  a  priori 


62  THE   THEORY  OF  RELATIVITY. 

improbable  in  the  conclusions  of  relativity  concerning  the 
nature  of  mass.  Therefore  if  we  find  satisfactory  grounds 
for  accepting  the  initial  postulates  of  relativity,  we  shall  not 
throw  these  postulates  overboard  because  of  the  strange  con- 
clusions concerning  mass  to  which  they  have  led  us. 

Now,  if  mass  is  merely  a  manifestation  of  energy  in  the 
form  of  a  strain  in  the  ether  it  would  follow  that  gravitation 
is  simply  an  interaction  among  these  several  strains.  A  strain 
principally  localized  in  one  place  would  have  lines  of  strain 
going  out  from  it  in  all  directions,  and  the  action  of  these  lines 
of  strain  upon  one  another  would  afford  the  effective  means 
by  which  gravitation  acts. 

§31.    THE  MASS  OF  LIGHT. 

From  some  results  in  the  preceding  discussion  it  has  appeared 
that  the  transverse  mass  of  a  body  is  merely  a  manifestation 
of  its  total  energy,  that  it  is  in  fact  a  measure  of  that  energy. 
It  is  then  natural  to  suppose,  on  the  other  hand,  that  anything 
which  possesses  energy  has  mass;  and  we  thus  conceive  of 
mass  and  energy  as  coextensive. 

Now  a  beam  of  light  possesses  energy;  whence  we  con- 
clude naturally  that  it  also  has  mass.  But  we  have  seen  that 
no  "  material  body  "  can  have  a  velocity  as  great  as  that  of 
light.  How  are  these  two  facts  to  be  reconciled?  If  we  define 
"  matter  "  as  that  which  possesses  mass  (and  this  is  probably 
the  best  definition)  we  shall,  as  we  have  seen,  perhaps  best 
be  able  to  represent  to  ourselves  the  nature  of  matter  if  we 
think  of  it  as  a  strain  in  the  ether.  Then  the  two  facts  which 
we  have  to  reconcile  would  be  entirely  consistent  if  we  suppose 
that  the  beam  of  light  sets  up  a  strain  in  the  ether  (whence 
its  mass)  but  that  this  strain  as  a  whole  is  not  propagated 
with  the  velocity  of  light.  In  fact,  if  it  moves  at  all  it  is 
probably  with  a  velocity  much  smaller  than  that  of  light. 


CHAPTER  VI. 
EXPERIMENTAL  VERIFICATION  OF   THE   THEORY. 

§32.    Two  METHODS  OF  VERIFICATION. 

THERE  are  at  least  two  ways  in  which  it  may  be  possible 
to  demonstrate  experimentally  the  accuracy  of  the  theory  of 
relativity. 

The  first  method  is  direct.  It  consists  in  the  proof  by 
experiment  of  the  postulates  on  which  the  theory  is  based. 
These  proved,  the  whole  theory  then  follows  by  logical  proc- 
esses alone.  In  Chapter  II  we  have  given  a  sufficient  discussion 
of  this  method. 

The  second  method  is  indirect;  it  may  be  described  as 
follows:  Among  the  consequences  of  the  theory  of  relativity 
seek  out  one  which  has  the  property  that  if  it  is  assumed  the 
postulates  of  relativity  may  themselves  then  be  deduced  by 
logical  processes  alone.  If  then  this  assumption  is  proved 
experimentally  this  is  sufficient  to  establish  the  postulates  of 
relativity,  and  hence  the  whole  theory.  Or,  one  may  find 
such  experimental  results  as  lead  to  all  the  essential  conclusions 
of  relativity,  whence  one  naturally  concludes  to  the  accuracy 
of  the  whole  theory.  A  discussion  of  proofs  of  this  kind  will 
be  given  in  this  chapter. 

This  indirect  method  of  proof  is  in  many  cases  open  to  an 
objection  of  a  kind  which  does  not  obtain  in  the  case  of  the 
direct  method  previously  mentioned.  In  the  indirect  method 
some  auxiliary  law,  as  for  instance  the  law  of  conservation  of 
electricity,  must  usually  be  employed  in  deducing  the  relativity 
postulates  or  essential  conclusions  from  the  new  assumption 
which  one  seeks  to  justify  by  experiment.  There  is  always 

63 


64  THE   THEORY   OF  RELATIVITY. 

the  possibility  that  the  auxiliary  law  itself  is  wrong;  and  con- 
sequently one's  confidence  in  the  accuracy  of  the  relativity 
postulates  as  thus  deduced  can  be  no  stronger  than  that  in 
the  truth  of  the  auxiliary  law.  The  same  objection  can  also 
be  raised  against  many  conclusions  which  we  are  accustomed 
to  accept  with  confidence. 

To  many  persons  it  appears  that  the  first  method  of  proof' 
mentioned  above  has  been  carried  out  successfully  and  satis- 
factorily. But  if  one  does  not  share  this  opinion  it  is  still 
legitimate  to  accept  the  theory  of  relativity  as  a  working 
hypothesis,  to  be  proved  or  disproved  by  future  experiment. 
It  is  an  historical  fact,  patent  to  every  student  of  scientific 
progress,  that  many  of  our  fundamental  laws  have  been  accepted 
in  just  this  way.  Take,  for  instance,  the  law  of  conservation 
of  energy.  There  is  no  experimental  demonstration  of  this 
law;  and  in  the  very  nature  of  things  it  is  hard  to  see  how  there 
could  be.  On  the  other  hand  it  is  at  variance  with  no  known 
experimental  fact.  Moreover,  it  furnishes  us  a  very  valuable 
means  of  systematizing  our  known  facts  and  representing  them 
to  our  minds  as  an  ordered  whole.  In  other  words,  it  is  the 
most  convenient  hypothesis  to  make  in  the  face  of  the  phenom- 
ena which  we  have  observed.  Similarly,  even  if  one  does  not 
believe  that  the  theory  of  relativity  has  been  conclusively 
demonstrated,  should  he  not  accept  that  theory  (tentatively 
at  least)  provided  it  furnishes  him  with  the  most  convenient 
means  of  representing  external  phenomena  to  his  mind? 

It  should  further  be  said  that  every  supposed  proof  of  the 
theory  of  relativity  is  of  such  character  that  objections  can  be 
raised  to  it;  likewise  every  supposed  disproof  of  the  theory 
is  in  the  same  state.  In  the  meantime,  though  we  cannot 
accept  the  theory  with  all  confidence,  we  can  at  least  use  its 
conclusions  to  suggest  experiments  which  otherwise  would  not 
have  been  conceived.  Therefore,  whether  true  or  false,  the 
theory  will  be  useful  in  the  advancement  of  science. 


EXPERIMENTAL  VERIFICATION  OF  THE  THEORY.  65 

§33.    LOGICAL  EQUIVALENTS  OF  THE  POSTULATES. 

In  every  body  of  doctrine  which  consists  of  a  finite  number 
of  postulates  and  their  logical  consequences  there  are  necessarily 
certain  theorems  which  have  the  following  fundamental  relation 
to  the  whole  body  of  doctrine:  By  means  of  one  of  these 
theorems  and  all  the  postulates  but  one  that  remaining  pos- 
tulate may  be  demonstrated.  That  is,  one  may  assume  such 
a  theorem  in  place  of  one  of  the  postulates  and  then  demonstrate 
that  postulate.  When  the  postulate  has  thus  been  proved  it 
may  be  used  in  argument  as  well  as  the  theorem  itself;  hence 
it  is  clear  that  all  of  the  consequences  which  were  obtained 
from  the  first  set  of  postulates  may  now  be  deduced  again, 
though  perhaps  in  a  somewhat  different  manner.  That  is, 
if  we  consider  the  whole  body  of  doctrine,  composed  of  pos- 
tulates and  theorems,  this  totality  is  the  same  in  the  two  cases. 
Two  sets  of  postulates  which  thus  give  rise  to  the  same  body 
of  doctrine  (consisting  of  postulates  and  theorems  together) 
are  said  to  be  logically  equivalent. 

The  problem  of  the  logical  equivalents  of  a  given  set  of 
postulates  is  readily  seen  to  be  an  important  one.  The  prin- 
cipal value  of  such  a  matter,  from  the  point  of  view  of  physical 
science,  consists  in  the  fact  that  it  affords  alternative  methods 
for  the  experimental  proof  or  disproof  of  a  theory  and  that 
it  emphasizes  in  an  effective  way  the  essential  difficulties  and 
limitations  of  such  experimental  verification  in  general. 

When  the  indirect  method  of  demonstration  described  in 
§32  is  carried  out  by  means  of  logical  equivalents  of  the  pos- 
tulates it  is  not  open  to  the  objection  mentioned  above.  Unfor- 
tunately, it  seems  to  be  difficult  to  carry  it  out  in  this  way, 
and  consequently  we  are  forced  to  a  method  of  procedure  less 
satisfactory,  at  least  from  the  point  of  view  of  logic. 

§34.    ESSENTIAL  EQUIVALENTS  OF  THE  POSTULATES. 

If  one  is  interested  in  the  theory  of  relativity  on  account 
of  its  significance  to  physical  science  it  is  unnecessary  to  have 
complete  logical  equivalents  of  the  postulates  in  order  to  justify 


66  THE   THEORY   OF   RELATIVITY. 

it.  All  that  is  essential  is  to  find  a  set  of  postulates,  experi- 
mentally demonstrable,  by  means  of  which  it  is  possible  to 
demonstrate  the  characteristic  conclusions  of  relativity  con- 
cerning the  relations  of  units  of  time  and  units  of  length  in 
two  systems  of  reference.*  Such  a  set  of  postulates  we  shall 
call  essential  equivalents  of  the  postulates  of  relativity.  The 
object  of  this  section  is  to  determine  essential  equivalents  of 
postulate  R,  that  is,  such  postulates  as  may  be  taken  in  con- 
nection with  postulates  M,  V,  L,  so  that  the  new  set  shall 
be  essentially  equivalent  to  M  ,  V,  L,  R. 

For  this  purpose  let  us  first  consider  the  relation  between 
the  transverse  mass  of  a  moving  body  and  its  mass  at  rest 
as  given  in  theorem  XI.  Let  us  suppose  that  this  theorem 
is  truef  (whether  proved  experimentally  or  otherwise);  and 
let  us  seek  its  consequences.  Suppose  that  the  experiment 
by  means  of  which  we  proved  theorem  XI  is  now  repeated. 
If  we  again  assume  the  law  of  conservation  of  momentum 
and  equate  the  two  observed  changes  in  momenta,  it  is  clear 
that  we  shall  have  a  relation  between  measurements  of  time  as 
carried  out  in  the  two  systems  of  reference,  and  that  this 
relation  will  be  precisely  the  same  as  in  the  usual  theory  of 
relativity.  Having  this  relation  concerning  time  units  we  can 
then  proceed  as  in  the  first  paragraph  in  §  25  to  derive  the 
usual  relations  between  units  of  lengths.  Hence  we  have  the 
following  result: 

THEOREM  XVI.  If  mo  and  t(mv)  have  the  same  meaning 
as  in  theorem  XI  and  if  for  any  particular  kind  of  matter  whatever 
we  have  the  relation 


then  this  fact  and  postulates  (MVLCi)  form  an  essential  equivalent 
of  postulates  (M  VLRCi)  . 

*  It  is  obvious  that  we  should  then  be  able  to  demonstrate  theorems  XI 
and  XII  concerning  the  mass  of  a  moving  body. 

t  All  that  is  essential  to  the  argument  is  the  truth  of  theorem  XI  for  a  particle 
of  matter  of  some  one  kind;  it  need  not  be  assumed  to  be  true  universally. 


EXPERIMENTAL   VERIFICATION   OF   THE   THEORY.  67 

Next,  let  us  suppose  that  for  some  particular  kind  of  matter 
we  have  the  relation 


where  t(mv)  and  l(mv)  denote  the  transverse  mass  and  the 
longitudinal  mass,  respectively,  as  in  §  24.  Then  repeat  the 
experiments  by  means  of  which  we  proved  theorem  XII.  As 
before  the  balls  will  appear  to  B  to  move  on  arcs  of  the  circle 


Suppose  that  to  A  they  appear  to  move  along  arcs  of  the 
ellipse* 

x  =  cos  0,     y  =  p  sin  0, 

where  p  is  a  constant  to  be  determined.  As  before,  without 
the  use  of  postulate  7?,  it  may  be  shown  that  to  A  the  periods 
will  be  the  same  in  the  two  cases.  Then  determine  the  periods 
as  in  the  preceding  discussion.  The  expression  for  the  periods 
will  contain  p;  in  fact  on  equating  them  we  shall  find 


But   mi=l(mv)    amd   m2  =  t(mv);    whence   on   account   of    the 
relation  between  l(mv)  and  t(m,)j  we  have  at  once 


This,  in  connection  with  postulate  L,  leads  readily  to  the  usual 
relations  concerning  the  units  of  length  in  two  systems  of 
reference.  Having  these  relations  of  length  units,  the  dimen- 
sional equation 


taken  in  connection  with  postulate  V  leads  at  once  to  the 
usual  relation  of  time  units,  provided  we  take  the  motion 
along  the  line  of  relative  motion  of  the  two  systems.  Hence 
we  have  the  following  theorem: 

*  Since  we  are  assuming  postulate  L  it  is  clear  that  the  path  must  be  of  this 
form. 


68  THE  THEORY  OF  RELATIVITY. 

THEOREM  XVII.  If  l(mD)  and  t(mv)  have  the  same  meaning 
as  in  theorems  XI  and  XII  and  if  for  any  particular  kind  of 
matter  whatever  we  have  the  relation 


then  this  fact  and  postulates  (MVLCiCz)  are  essential  equivalents 
of  postulates  (MVLRCiC2). 

§35.    THE  BUCHERER  EXPERIMENT. 

Our  postulates  V,  L,  C\  have  been  universally  accepted 
as  part  of  the  basis  of  the  classical  mechanics.  Many  persons 
have  found  no  difficulty  in  accepting  postulate  M;  certain  it 
is  at  least  that  we  have  absolutely  no  evidence  to  contradict 
it.  We  have  seen  in  theorem  XVI  that  these  four  postulates, 
taken  in  connection  with  the  formula  for  transverse  mass, 
form  an  essential  equivalent  of  (MVLRCi)',  in  other  words, 
the  experimental  demonstration  of  the  formula  for  transverse 
mass  carries  with  it  the  experimental  proof  of  the  theory  of 
relativity,  provided  that  postulates  (MVLCi)  are  accepted  as 
experimentally  proved. 

Bucherer  (Annalen  der  Physik,  ser.  4,  vol.  28,  pp.  513-536) 
has  carried  out  some  investigations  which  have  been  supposed 
to  furnish  this  experimental  verification  for  the  formula  of 
transverse  mass,  and  hence  for  the  whole  theory  of  relativity. 
In  order  to  draw  this  conclusion  from  Bucherer's  direct  results 
it  is  necessary  to  make  use  of  a  law  which  we  have  not  yet 
employed,  namely,  the  law  of  conservation  of  electricity  which 
we  have  stated  as  postulate  €3.  Since  this  law  has  customarily 
been  accepted,  we  shall  conclude  that  we  have  in  Bucherer's 
results  a  partial  experimental  confirmation  of  the  theory  of 
relativity. 

Bucherer's  investigations  have  to  do  with  the  mass  of  a 
moving  electron.  There  seems  to  be  no  means  at  hand  for  a 
direct  measurement  of  this  mass,  and  Bucherer  resorted  to  the 
expedient  of  determining  the  ratio  of  charge  to  mass.  Let 
us  denote  the  charge  by  e,  which  we  suppose  to  be  constant, 


EXPERIMENTAL  VERIFICATION   OF   THE   THEORY.  69 

in  accordance  with  postulate  Ca-  As  before  let  mo  and  t(mv) 
denote  the  mass  at  rest  and  the  transverse  mass  when  moving 
with  velocity  v,  of  the  electron  in  consideration.  Bucherer's 
experiments  were  carried  out  to  determine  the  relation  which 
exists  between  e/mQ  and  e/t(m,}.  The  measurements  agreed 
in  a  remarkable  way,  not  only  as  to  general  characteristics 
but  also  as  to  exact  numerical  results,  with  the  formula* 


_e eVi  -  ft2 

t(mv)  ~      m0 

Taking  this  formula  as  thus  experimentally  demonstrated  we 
have  at  once  our  fundamental  relation  for  transverse  mass: 

Vi  —  $2-t(mv)=mo. 

From  this  it  follows  that  the  experimental  demonstration 
of  the  theory  of  relativity  is  complete  when  we  have  proved 
M,  V,  L,  Ci  and  Ca,  provided  that  one  accepts  Bucherer's 
proof  of  the  above  relation  between  e/mo  and  e/t(mv).  That 
is,  the  essentials  of  the  theory  of  relativity  flow  from  principles 
for  each  of  which  there  is  strong  experimental  confirmation.  This 
important  conclusion  has  often  been  pointed  out. 

To  the  present  writer,  however,  it  seems  that  one  point 
especially  should  be  subjected  to  further  examination.  Is  it 
in  fact  true  that  the  charge  of  a  moving  electron  is  independent 
of  the  velocity  with  which  it  moves?  Let  eo  be  the  charge 
of  the  electron  when  at  rest  and  denote  by  t(ev)  its  apparent 
charge  when  in  motion  with  velocity  v,  the  charge  being  measured 
by  means  of  tests  in  which  the  line  of  action  is  perpendicular 

*  As  a  matter  of  fact  Bucherer  did  not  measure  the  ratio  e/mo.  Instead  of 
this  he  considered  the  ratio  e/t(mv]  for  a  considerable  range  of  values  for  v  and 
noticed  that  its  value  always  agreed  with  the  formula  e/t(mv)=k\/i  —  p-,  where 
k  is  a  constant.  It  appears  natural,  then,  to  assume  that  mo=e/k,  whence  one 
has  the  formula  in  the  text.  It  should  be  emphasized  that  this  assumption  is 
necessary  in  order  that  the  Bucherer  results  may  be  associated  with  our  theorem 
as  in  the  text,  and  consequently  the  conclusions  there  reached  can  be  accepted 
with  no  stronger  confidence  than  that  which  one  has  in  the  accuracy  of  the  above 
assumption.  See  the  next  section  where  a  possible  means  of  experimental  veri- 
fication of  the  theory  of  relativity  is  suggested  which  does  not  depend  on  this 
assumption  for  its  validity. 


70  THE   THEORY   OF   RELATIVITY. 

to  the  line  of  motion  of  the  charge.  In  the  above  discussion 
we  have  assumed,  in  accordance  with  the  usual  practice,  that 
€Q  =  t(ev).  Suppose  however  that  the  true  relation  were  different 
from  this,  that,  in  fact,  we  have 


-e2; 


then  Bucherer's  experiment  would  lead  to  the  conclusion  that 
t(mv)  =  mo,  and  thus  the  whole  theory  of  relativity  would  be 
overturned.  Furthermore,  if  any  relation  other  than  eo  =  t(ec) 
is  the  true  one,  some  modification  at  least  of  the  theory  of 
relativity  would  have  to  be  made  or  else  one  would  have  to 
give  up  postulate  C\  which  asserts  the  law  of  conservation  of 
momentum.  This  result  emphasizes  the  great  importance  of 
the  question  of  the  constancy  of  electric  charge  on  the  electron. 
We  shall  treat  this  matter  further  in  the  next  section. 

§  36.    ANOTHER  MEANS  FOR  THE  EXPERIMENTAL  VERIFICATION 
OF  THE  THEORY  OF  RELATIVITY. 

Just  as  theorem  XVI  was  used  for  the  theoretical  basis  of 
Bucherer's  (partial)  experimental  demonstration  of  the  theory 
of  relativity  so  theorem  XVII  may  be  employed  as  the  theo- 
retical basis  of  a  new  experimental  investigation  which  has 
not  yet  been  carried  out,  one  which  bears  the  same  essential 
relation  as  that  of  Bucherer  to  the  confirmation  or  disproof 
of  the  entire  theory  of  relativity.  The  object  of  this  section 
is  to  indicate  the  nature  of  this  experiment. 

Let  eo  denote  the  charge  of  an  electron  when  at  rest  with 
respect  to  a  given  system  of  reference.  When  it  is  in  motion 
with  a  velocity  v  let  t(ev)  and  l(ev)  be  the  apparent  charge  when 
measured  by  means  of  tests  whose  lines  of  action  are  perpen- 
dicular and  parallel,  respectively,  to  the  line  of  motion  of  the 
electron. 

If  we  employ  postulate  Cs  we  conclude  that  eo  =  t(ev)=l(ev). 
We  shall  first  assume  the  truth  of  one  of  these  relations,  namely, 
t(ev)  =l(ev),  and  we  shall  denote  the  common  value  of  these  two 
quantities  by  e.  Now  let  us  suppose  that  some  means  are 


EXPERIMENTAL  VERIFICATION   OF   THE   THEORY.  71 

found  for  measuring  both  the  quantities  e/t(mc)  and  e/l(mv), 
where  t(mv)  and  l(mv)  denote  as  usual  the  transverse  mass 
and  the  longitudinal  mass  respectively  of  the  moving  electron, 
whose  velocity  is  v.  Bucherer's  methods  furnish  a  means  of 
measuring  the  first  of  these  ratios;  it  will  be  necessary  to  devise 
a  way  to  determine  the  value  of  the  second  ratio. 

Or,  instead  of  finding  a  means  of  measuring  the  two  quan- 
tities e/t(mv)  and  e/l(m,)  it  will  be  sufficient  if  one  determines 
only  their  ratio,  as  will  be  obvious  from  the  discussion  following. 

Suppose  now  that  we  find  the  relation  predicted  by  the 
theory  of  relativity: 


This  equation  leads  to  the  relation  t(mv)  =  (i  —  g2)  •  l(mv)  .  Accord- 
ing to  theorem  XVII  this  would  give  a  new  experimental 
confirmation  of  the  theory  of  relativity.  The  importance  of 
such  a  result  is  apparent. 

But  we  should  also  have  more  than  this.  Having  now 
concluded  that  the  theory  of  relativity  is  confirmed  and  this 
result  having  been  reached  without  the  use  of  a  relation  between 
CQ  and  t(ev),  we  may  now  use  the  experiment  of  Bucherer  to 
draw  further  conclusions  concerning  electric  charges  in  motion. 
In  particular,  it  is  obvious  that  we  should  have  a  proof  of  the 
fundamental  relation 


That  is  to  say,  having  assumed  that  t(ec)  and  l(ev)  are  equal 
we  conclude  further  on  direct  experimental  evidence  that  each 
of  these  is  equal  to  eo.  Now  it  is  difficult  to  conceive  how 
t(ev)  and  l(ev)  could  be  different,  for  this  would  imply  that 
the  notion  of  electric  charge  is  in  need  of  essential  modification. 
In  fact,  if  the  charged  body  is  moving,  the  notion  of  charge 
would  be  indefinite  in  meaning  until  we  had  assigned  the 
direction  along  which  such  charge  is  to  be  measured.  Thus, 
if  the  experiment  should  turn  out  as  surmised  above,  we  should 
not  only  have  the  strongest  sort  of  experimental  confirmation 
of  the  theory  of  relativity  but  we  should  also  have  a  valuable 


72  THE  THEORY  OF  RELATIVITY. 

verification  of  the  fact  that  an  electric  charge  does  not  vary 
in  amount  with  the  velocity  of  the  body  which  carries  it. 

Suppose,  on  the  other  hand,  that  we  make  no  assumption 
concerning  the  relation  of  t(et)  and  l(e,)  or  of  t(mc)  and  l(mv). 
On  carrying  out  the  experiments  a  relation  of  the  form 

t(e,)        l(e,) 


will  be  obtained  where  k  is  a  constant  or  a  variable  depending 
on  v.  If  it  is  found  that  k  is  different  from  unity  we  shall 
be  forced  to  the  conclusion  that  either  our  conception  of  mass 
in  the  classical  mechanics  or  our  conception  of  charge  in  the 
classical  electrical  theory  is  in  need  of  essential  modification. 
Again,  if  k  =  i  and  if  we  assume,  as  is  natural,  that  t(ev)=l(ev). 
then  we  have  an  experimental  disproof  of  the  theory  of  rela- 
tivity. In  fact  we  have  such  a  disproof  unless  &  =  i/(i  —  p2), 
provided  of  course  that  we  assume  t(ev)  =l(ev). 

From  these  remarks  it  is  obvious  that,  whatever  may  be 
the  result  of  the  experiments,  they  will  certainly  lead  to  important 
conclusions  of  a  fundamental  nature;  that  is,  we  have  here 
a  crucial  experiment,  one  that  cannot  fail  to  lead  somewhither. 
It  is  to  be  hoped  that  some  laboratory  worker  will  soon  perform 
the  requisite  experiments;  the  writer,  who  is  a  mathematician, 
can  only  regret  that  he  cannot  conveniently  carry  out  the 
work  himself. 

A  variation  of  the  experiment  of  Bucherer  would  seem  to 
be  sufficient  for  the  purpose  here.  Bucherer's  results  were 
obtained  by  subjecting  the  moving  electron  to  a  magnetic 
field  and  also  to  an  electric  field  each  at  right  angles  to  the 
line  of  motion.  A  variation  of  the  direction  of  these  fields 
relative  to  the  line  of  motion  of  the  electron  would  probably 
afford  a  means  of  making  the  necessary  measurements  for 
the  experimental  proof  of  the  relations  requisite  for  use  in 
the  preceding  discussion. 


CHAPTER  VII. 

THE  GENERALIZED  THEORY  OF  RELATIVITY. 
§  37.  SUMMARY  OF  RESULTS  FROM  PREVIOUS  CHAPTERS. 

WE  have  seen  (in  §  20)  that  the  restricted  theory  of  rela- 
tivity, as  developed  in  the  preceding  pages,  calls  for  a  trans- 
formation of  a  remarkable  kind  between  the  coordinates  of 
two  systems  moving  with  a  uniform  velocity  v  relatively  to  each 
other.  If  x,  y,  z,  t  are  the  space-time  coordinates  of  a  system  5 
and  x',  y',  z'  ,  t'  are  the  space-time  coordinates  of  a  system  5", 
related  to  S  as  in  §  20,  then  we  have 

(i)  t'  =  y(t-^x),    x*  =  y(x-vt),    y'  '  =  y,    zf  =  z, 

where 


and  c  is  the  velocity  of  light.  If  these  equations  are  solved  for 
x,  y,  z,  t  in  terms  of  x',  y',  z'  ',  /',  the  resulting  equations  are  what 
those  in  (i)  become  on  interchanging  the  primed  with  the 
unprimed  coordinates  and  replacing  v  by  —v.  The  transforma- 
tions thus  have  a  remarkable  symmetry. 
From  the  first  two  equations  in  (i)  we  have 

dx'       dx  —  vdt 


dt'      dt-vdx/c* 
or, 

,       u— v 


(2)  «' 


I-11V/C2' 


if  u'  =  dx'/dt'  and  u  =  dx/dt,  these  denoting  the  velocities  of 
motion  in  the  ^-direction  of  the  coordinate  axes  in  each  system 

73 


74  THE  THEORY  OF  RELATIVITY. 

with  respect  to  that  system.  Formula  (2)  states  the  principle 
of  addition  of  velocities,  from  it,  as  we  have  seen  (in  §  21),  it 
follows  that  the  usual  law  of  addition  of  velocities  is  not  main- 
tained. 

But  Robb  has  pointed  out  that  if  we  introduce  the  notion 
of  rapidity  of  motion  and  define  the  measure  of  rapidity  of 
motion  with  a  velocity  v  to  be  tanh"1  (v/c),  then  we  do  have 
for  rapidities  a  simple  law  of  addition,  namely,  that  obtained 
readily  from  equation  (2)  and  expressible  in  the  form 

-'(' 

V 

Again,  wre  have  found  (in  §  24)  that  the  mass  of  a  body  is 
dependent  upon  its  velocity  relative  to  the  observer's  system, 
and  in  fact  that  the  transverse  mass  t(mv)  of  a  body  in  motion 
with  a  uniform  velocity  v  relative  to  the  system  is  expressed  in 
terms  of  the  mass  mo  of  the  body  at  rest  by  the  formula 


The  formula  for  longitudinal  mass  is  also  given  in  §  24;  since 
we  have  no  further  need  for  the  conception  of  longitudinal  mass 
that  formula  will  not  be  repeated  here.  Hereafter  we  shall 
use  the  word  "mass"  to  denote  what  we  have  heretofore  called 
transverse  mass. 

In  §  26  we  have  seen  that  mass  and  energy  are  interchange- 
able in  the  sense  that  the  measure  of  each  may  be  expressed 
directly  in  terms  of  the  measure  of  the  other.  A  consequence 
of  this  is  that  we  may  use  in  our  equations  energy  and  not 
mass  or  mass  and  not  energy  instead  of  both  mass  and  energy, 
if  it  should  turn  out  that  such  a  thing  shall  serve  our  convenience. 

In  the  generalized  theory  of  relativity  it  can  hardly  be  said 
that  the  notion  of  force  enters  at  all;  consequently  we  shall 
not  need  to  employ  in  this  chapter  the  formula  (of  §  25)  for 
the  transformation  of  force  in  passing  from  one  system  of  refer- 
ence to  another. 


THE  GENERALIZED  THEORY  OF  RELATIVITY.  75 

§  38.  TRANSFORMATIONS  IN  SPACE  OF  FOUR  DIMENSIONS. 

Following  a  suggestion  of  Minkowski's,  we  may  look 
upon  the  transformation  (i)  as  in  a  certain  sense  the  trans- 
formation due  to  a  rotation  of  axes  in  a  space-time  extension 
of  four  dimensions.  For  the  purpose  of  viewing  it  in  this  light 
let  us  assume  that  the  units  are  so  chosen  that  the  velocity  c 
of  light  is  unity  and  let  us  make  the  imaginary  transformation 


Then  we  have  7  =  cos  6  and  equations  (i)  reduce  readily  to  the 
form 

(3)     r  =r  cos  8—  x  sin  6,    x'  =  x  cos  0+r  sin  6,    yf  =  y,    z'  =  z. 

This  represents  a  rotation  of  the  axes  of  coordinates  through  an 
imaginary  angle  6  in  the  xr-plane. 

In  the  Newtonian  mechanics  the  laws  of  nature  are  assumed 
to  be  invariant  with  respect  to  a  change  in  the  orientation  of 
axes  in  the  :ryz-space.  In  the  theory  of  relativity  this  prin- 
ciple is  extended  to  the  four-dimensional  space-  time  xyzr- 
extension  ;  and  it  is  therefore  assumed  that  this  four-dimensional 
extension  is  completely  isotropic.  From  this  point  of  view  the 
conspiracy  of  nature  to  prevent  our  measuring  the  velocity  of 
our  system  through  space  disappears;  there  is  nothing  to 
conceal.  Space-time  extension  being  isotropic,  there  is  no 
variation  of  properties  in  different  directions  and  hence  nothing 
for  us  to  detect;  no  one  orientation  is  more  fundamental  than 
another. 

One  consequence  of  this  is  that  we  cannot  pick  out  one 
direction  as  the  absolute  time  any  more  than  we  can  pick  out 
one  direction  as  the  absolute  vertical.  As  Minkowski  has  said: 
"Henceforth  space  and  time  in  themselves  vanish  to  shadows, 
and  only  a  kind  of  union  of  the  two  preserves  an  independent 
existence."  Unfortunately  the  simplicity  of  the  conception  of  a 
four-dimensional  space-time  extension  and  the  interpretation 
of  our  transformation  (i)  as  due  to  a  rotation  in  it  are  marred 


76  THE  THEORY  OF  RELATIVITY. 

by  the  fact  that  nature  makes  a  distinction  of  such  sort  that 
we  have  to  employ  the  imaginary  time  variable  r  in  order  to 
exhibit  our  transformation  as  having  the  properties  of  a  rotation 
in  a  space  of  four  dimensions. 

If  we  use  ids  to  represent  the  element  of  "length"  in  our 
space-time  extension,  that  is,  the  interval  between  two  point- 
events,  its  value  will  be  given  by  the  relation 

(4)  -  ds2  =  dx2+dy2+dz2+dr2. 

It  is  easy  to  see  that  ds  is  invariant  with  respect  to  a  rotation 
of  axes  and  in  particular  with  respect  to  the  rotation  defined 
by  equations  (3).  If  we  choose  axes  moving  with  the  particle 
we  have,  dx  =  dy  =  dz  =  o,  so  that  ds2=—dr2  or  ds  =  dt',  this 
explains  the  choice  of  sign  in  the  first  member  of  (4) ,  the  purpose 
being  to  secure  in  5  a  sort  of  time  variable. 

It  can  be  shown  that  the  whole  restricted  theory  of  relativity, 
as  developed  in  the  preceding  chapters,  is  summed  up  in  the 
conclusion  that  ds  is  invariant;  and  from  the  hypothesis  of  the 
in  variance  of  ds  one  can  deduce  the  transformation  equations  (i), 
and  hence  the  other  fundamental  results  of  the  theory. 

We  have  used  the  rysr-extension  for  convenience  in  deriving 
certain  geometric  properties  of  the  transformation  (i).  It  is 
clear  that  we  may  likewise  look  upon  x,  y,  z,  t  as  coordinates 
of  a  point  in  the  real  space-time  ^/-extension  of  four  dimen- 
sions. Let  us  consider  a  little  more  closely  this  space-time 
extension.  Each  point  P  of  this  four-dimensional  extension 
represents  both  a  definite  place  A  in  the  usual  space  of  three 
dimensions  and  a  definite  moment  of  time  /  at  which  this  place 
A  is  to  be  considered.  If  P  refers  to  a  material  point  it  shows 
the  time  t  at  which  this  point  is  found  at  the  place  A.  In  the 
course  of  time  the  material  point  is  represented  every  moment 
by  a  new  point  P  of  the  space- time  extension;  all  these  points 
P  lie  on  a  world  line  which  represents  completely  once  for  all 
the  state  of  motion  or  of  rest  of  the  material  point  for  all  time. 
In  the  same  way  we  may  speak  of  the  world-line  of  any  event 
in  nature.  An  intersection  of  two  world-lines  indicates  that 
the  two  objects  to  which  they  belong  meet  at  a  certain  moment, 


THE  GENERALIZED  THEORY  OF  RELATIVITY.  77 

that  a  "coincidence"  takes  place.  Similarly,  one  may  repre- 
sent statically  in  a  space  of  three  dimensions  the  kinetics  of  a 
body  moving  in  a  plane;  a  clear  picture  of  this  special  case 
will  assist  one  greatly  in  grasping  the  more  general  considera- 
tions for  motion  in  a  space  of  three  dimensions. 

Now  Einstein  has  remarked  that  the  only  things  which  we 
can  observe  and  measure  among  events  in  nature  are  these 
coincidences,  the  intersections  of  these  world-lines,  and  that  it 
is  with  these  alone  that  our  theories  are  essentially  concerned. 
From  this  it  follows  that  the  results  of  all  observations  may 
be  represented  by  world-lines  in  a  four-dimensional  extension — let 
us  say  by  means  of  a  field-figure — and  that  the  only  things  directly 
observed  are  the  intersections  of  these  lines  one  with  another. 

In  our  statement  of  the  laws  of  nature  we  shall  have  to  attend 
only  to  the  intersections  of  the  world-lines;  and  hence  we  shall 
have  a  great  liberty  in  the  construction  of  the  field-figures.  We 
may  vary  this  construction  in  any  way  we  please,  provided 
only  that  we  do  not  disturb  the  order  of  the  intersections  of 
the  world-lines.  As  the  number  of  observed  intersections 
increases  we  are  more  restricted  in  this  freedom,  but  only  in 
the  way  of  a  finite  number  of  added  restrictions  to  the  infinitude 
of  possible  changes  inherent  in  the  nature  of  the  process. 
Even  if  all  the  intersections  in  nature  were  known  there  would 
still  be  great  freedom  in  the  construction  of  the  field-figures. 
If  two  persons  independently  represent  the  same  observations 
by  means  of  world-lines  their  field-figures  will  probably  be  quite 
different  and  in  fact  will  probably  agree  only  as  to  the  order  of 
the  points  of  intersection,  all  other  properties  of  the  two  figures 
being  different;  and  indeed  these  figures  will  quite  as  well 
represent  the  observed  facts  after  being  deformed  in  any  way 
provided  that  there  is  no  break  in  continuity  during  the  process 
of  deformation. 

One  consequence  of  these  considerations,  which  is  important 
for  our  purposes,  is  that  there  is  a  great  freedom  of  choice  of 
coordinate  systems  in  reducing  the  observed  laws  to  analytical 
form  and  that  the  essential  laws  may  be  represented  quite  as 


78  THE  THEORY  OF  RELATIVITY. 

well  on  the  basis  of  one  of  these  systems  as  on  that  of  another. 
For  a  long  time  it  has  been  customary  to  introduce  various 
types  of  curvilinear  coordinates  or  of  moving  axes  for  the  study 
of  particular  problems  in  physics,  but  it  has  usually  not  been 
forgotten  even  for  a  moment  that  these  coordinate  systems 
were  curvilinear  or  were  in  motion.  But  there  is  at  least  one 
important  and  instructive  case  in  which  a  simple  means  has 
been  found  for  ignoring  the  peculiarity  of  the  axes  during  the 
mathematical  investigation.  This  is  the  case  of  rotating  axes. 

In  many  dynamical  systems,  some  part  of  the  system  is 
compelled  to  rotate  with  constant  angular  velocity  co  round  a 
given  fixed  axis.  The  motion  of  a  bead  on  a  rotating  twisted 
wire  furnishes  a  simple  example.  The  system  might  be  treated 
by  a  direct  application  of  Lagrange's  equations;  but  it  is  often 
more  convenient  to  use  a  principle  which  reduces  the  con- 
sideration of  systems  of  this  kind  to  that  of  systems  in  which 
no  rotation  takes  place.  When  one  develops  the  general 
differential  equations  of  motion  of  such  a  system  (as  in  Whitta- 
ker's  Analytical  Dynamics,  second  edition,  pp.  40-41,  for  in- 
stance) it  is  seen  that  the  motion  is  the  same  as  if  the  prescribed 
angular  velocity  were  zero  and  the  potential  energy  contained 
an  additional  term  —%2mr2u>2,  where  m  denotes  the  mass  and 
r  the  distance  of  a  particle  from  the  axis  of  rotation.  Thus,  by 
modifying  the  potential  energy,  we  may  replace  the  consideration 
of  a  system  which  is  constrained  to  rotate  uniformly  about 
an  axis  by  that  of  a  system  for  which  this  rotation  does  not 
take  place.  The  imaginary  forces  which  are  introduced  in  this 
way  to  represent  the  acceleration  effect  of  the  enforced  rotation 
are  often  called  centrifugal  forces. 

Now  let  us  suppose  that  an  observer  is  stationed  on  such  a 
rotating  system  and  that  he  is  shut  off  from  the  observation 
of  things  external  to  his  system  in  such  a  way  that  he  is  quite 
unaware  of  the  fact  that  it  is  rotating.  The  fictitious  term 
which  we  have  just  supposed  to  be  added  to  the  potential  energy 
to  account  mathematically  for  the  rotation  would  not  be 
fictitious  for  him  but  would  represent  an  existing  part  of  the 
actual  potential.  He  would  not  unnaturally  take  it  to  be  due 


THE  GENERALIZED  THEORY  OF  RELATIVITY.  79 

to  a  part  of  his  gravitational  field.  At  any  rate  he  would 
have  no  means  to  distinguish  it  from  a  potential  due  to  a  changing 
component  of  the  gravitational  field.  To  him  then  this  cen- 
trifugal force  would  seem  to  be  a  real  thing;  but  to  us,  who 
look  upon  his  system  from  the  outside,  this  centrifugal  force 
appears  to  be  fictitious. 

Now  suppose  that  a  ray  of  light  passes  across  this  observer's 
rotating  system.  We  who  look  upon  his  system  from  the  out- 
side will  see  that  the  ray  passes  in  a  straight  line.  But  to  him, 
on  account  of  the  unobserved  rotation  of  his  system,  it  will 
appear  to  move  on  a  curved  path.  If  he  sets  out  to  investigate 
the  curvature  of  this  path  he  will  find  that  it  depends  upon  that 
element  of  his  gravitational  field  which  we  consider  to  be  fictitious 
and  he  will  conclude  that  the  ray  of  light  is  bent  in  its  course  by 
the  gravitational  field,  or  at  least  by  a  certain  part  of  it. 

We  shall  not  pursue  this  further  in  the  loose  way  of  this 
section  but  shall  leave  it  to  be  taken  up  again  more  rigorously 
when  we  have  prepared  suitable  mathematical  machinery  for 
dealing  with  it. 

§  39.  THE  PRINCIPLE  OF  EQUIVALENCE. 

There  are  no  coordinate  axes  in  nature;  these  are  intro- 
duced by  us  for  convenience  in  the  analytical  representation 
of  phenomena.  When  we  enunciate  the  laws  of  mechanics 
and  electrodynamics  with  respect  to  "unaccelerated  rectangular 
axes,"  or  "Galilean  axes"  as  they  are  sometimes  called,  the  only 
definition  which  we  can  give  of  these  axes  is  that  they  are  the 
axes  with  respect  to  which  the  laws  may  be  enunciated  cor- 
rectly in  the  form  in  which  we  state  them.  We  cannot  recognize 
such  axes  intuitively.  One  fundamental  and  central  purpose  of 
the  generalized  theory  of  relativity  is  to  restate  the  laws  of  nature 
in  such  a  form  that  the  statement  shall  be  independent  of  the  sys- 
tem of  coordinates  and  hence  be  equally  applicable  to  all  systems. 

When  we  introduced  the  centrifugal  force  in  the  preceding 
section  and  then  looked  upon  the  rotating  system  of  reference 
as  stationary  so  that  moving  bodies  were  acted  upon  by  a 


80  THE  THEORY  OF  RELATIVITY. 

fictitious  force,  we  saw  that  the  changed  point  of  view  gave  us  a 
space  in  which  all  paths  (even  that  of  light)  were  deformed  in  a 
purely  geometric  way.  Everything  was  acted  upon  in  the 
same  way  by  this  fictitious  force.  This  property  is  also  shared 
by  the  force  of  gravitation;  the  gravitational  field  produces  an 
acceleration  which  is  independent  of  the  nature  or  the  mass  of 
the  body  acted  upon.  This  has  led  to  the  hypothesis  that  the 
force  of  gravitation  may  be,  so  far  as  we  can  observe,  of  essen- 
tially the  same  natuje  as  the  centrifugal  or  geometrical  forces 
introduced  by  the  choice  of  coordinates. 

This  hypothesis  has  been  framed  by  Einstein  into  his  now 
celebrated  Principle  of  Equivalence.  It  may  be  enunciated  as 
follows:  A  gravitational  field  of  force  is  exactly  equivalent 
to  a  field  of  force  introduced  by  a  transformation  of  the  coor- 
dinates of  reference  so  that  we  cannot  by  any  possible  experiment 
distinguish  between  them. 

Eddington  in  his  report  to  the  Physical  Society  of  London 
on  "The  Relativity  Theory  of  Gravitation"  has  insisted  on 
a  precise  criterion  for  the  cases  in  which  the  principle  of  equiva- 
lence is  assumed  to  apply.  He  formulates  it  as  follows:  The 
laws,  relating  to  phenomena  in  a  geometric  field  of  force,  which 
depend  on  the  coefficients  g  of  the  next  section  and  their  first 
derivatives,  will  also  hold  in  a  permanent  gravitational  field, 
namely,  one  that  cannot  be  entirely  removed  merely  by  a  change 
of  axes;  but  laws  which  depend  on  the  second  or  higher  deriv- 
atives of  the  g's  will  not  necessarily  have  this  universality. 

It  will  be  observed  that  this  contention  rests  upon  an 
implied  limitation  of  the  principle  of  equivalence.  It  is  argued 
that  there  is  such  a  thing  as  a  natural  or  permanent  gravita- 
tional field  of  force  which  cannot  be  altogether  transformed 
away.  There  is  a  tacit  agreement  that  a  natural  gravitational 
field  of  force  exists  even  though  it  may  be  altogether  impossible 
for  us  to  distinguish  between  it  and  the  fictitious  geometrical 
or  centrifugal  forces  due  to  the  choice  of  coordinate  axes. 

Transformations  exist  which  remove  the  gravitational  field 
at  a  point;  but  we  cannot  find  any  transformation  which  will 
remove  the  gravitational  field  throughout  a  finite  region. 


THE  GENERALIZED  THEORY  OF  RELATIVITY.  81 

Since  we  cannot  distinguish  between  the  fictitious  and  the 
permanent  forces  we  shall  have  no  means  in  general  of  selecting 
any  particular  system  of  coordinates  as  fundamental.  We 
must  therefore  state  the  laws  of  nature  in  a  form  which  is 
quite  independent  of  the  choice  of  axes. 

§  40.  GENERAL  TRANSFORMATION  OF  AXES. 

If  we  seek  a  suitable  analytical  expression  of  the  freedom 
which  we  have  seen  to  exist  in  our  construction  of  the  field- 
figures  of  natural  phenomena  we  shall  be  led  to  certain  con- 
siderations which  are  of  essential  importance  for  our  purposes. 
If  we  represent  in  terms  of  a  given  set  x,  y,  z,  t  of  space-time 
coordinates  the  configuration  of  a  system  of  world-lines  and  in 
terms  of  a  set  x'  ',  y',  z',  tr  the  configuration  of  another  system 
of  world-lines,  it  is  not  difficult  to  see  that  the  two  systems 
of  world-lines  will  have  corresponding  intersections  and  these 
arranged  in  the  same  relative  order  when  and  only  when  there 
exists  a  single-valued  continuous  transformation  from  the 
coordinates  of  each  system  to  those  of  the  other. 

Since  events  do  not  presuppose  any  particular  system  of 
coordinates  and  the  space-time  scaffolding  is  introduced  by  us 
merely  for  our  convenience,  it  is  desirable  to  have  our  laws  of 
nature  expressed  so  as  to  be  quite  independent  of  the  system 
of  coordinates  employed.  Let  us  see  what  this  amounts  to  in 
the  case  of  the  element  ds  whose  value  in  the  absence  of  a 
gravitational  field  is  given,  as  we  have  seen,  by  the  equation, 

(5)  ds*  =  -dx2-dy2-dz2+dt2. 

Let  us  introduce  new  coordinates  xi,  X2,  #3,  x±  by  means  of 
transformations  of  the  kind  just  mentioned;  they  are  given  by 
equations  of  the  form, 

X=fi(Xi,  X2,  #3,  #4),         y=/2(Xi,  X2)  XS,  #4),  CtC. 

Then  we  have 


(6)  dx=  <dXl  +     -dX2+-  dxs+lXi,  etc. 

8*1  3*2  C*3  C*4 


82  THE  THEORY  OF  RELATIVITY. 

Putting  such  values  of  dx,  dy,  dz,  dt  into  equation  (5),  we 
have  a  relation  which  may  be  written  in  the  form 

(7)          ds2  = 


where  the  g's  are  readily  computed  functions  of  the  coordinates 
xi,  X2,  xs,  #4,  depending  on  the  functions  /i,  /2,  fs,  /4  of  the 
transformation.  We  shall  henceforth  assume  that  ds  is  denned 
by  an  equation  of  the  form  (7),  without  reference  to  whether 
that  equation  is  derived  from  (5)  or  by  other  means. 

If  we  take  the  particular  transformation  of  rotating  axes 


x  =  xi  cos  co#4  —  x2  sn  ux±,  y  =  xi  sn  co#4#2  cos 

we  obtain  readily  the  relation. 

(8)  ds2=  -dx12-dx22-dx32+[i-u2(x12+X22)]dx42 


By  comparing  this  with  (7)  we  obtain  the  values  of  the  g's 
for  this  system  of  coordinates.     In  particular  we  have, 


where  fi  is  the  potential  of  the  centrifugal  force. 

From  this  it  follows  that  the  coefficient  #44  may  be  regarded 
as  a  potential;  and  this  conception  is  extended  so  that  all  the 
the  coefficients  g  are  regarded  as  components  of  a  generalized 
potential  of  the  field  of  force.  It  is  unnecessary  and  indeed 
experimentally  impossible,  according  to  the  theory  of  relativity, 
to  distingush  between  the  portion  of  the  g's  arising  from  the 
choice  of  coordinates  and  that  arising  from  the  so-called  natural 
or  permanent  gravitational  field.  We  shall  usually  speak  of 
the  entire  field  as  gravitational  and  shall  say  that  a  gravitational 
field  is  specified  by  a  set  of  values  of  the  g's  whatever  their  source. 

These  coefficients  g  may  be  looked  upon  in  two  ways.  In 
one  aspect  they  may  be  thought  of  as  expressing  the  metrical 
properties  of  the  coordinates;  and  this  is  the  orthodox  stand- 
point of  the  theory  of  relativity;  it  is  attained  by  banishing 


THE  GENERALIZED  THEORY  OF  RELATIVITY.  83 

almost  or  quite  entirely  the  notion  of  gravitational  force,  this 
exclusion  being  based  on  the  fact  that  if  there  are  such  things 
as  actual  gravitational  forces  we  cannot  distinguish  them  from 
fictitious  geometrical  forces. 

The  values  of  the  g's  given  by  equation  (5),  namely,  gn  =  - 1, 
£22= -i,  £33= -i,  £44  =  1,  gtj  =  o,  when  i^j,  are  those  which 
obtain  in  the  absence  of  a  gravitational  field  and  for  a  suitable 
choice  of  reference  system.  We  may  call  them  the  Galilean 
values  of  the  g's.  When  the  coordinates  can  be  chosen  so  that 
the  g's  have  these  values  we  may  regard  coordinates  so  chosen 
as  fundamental;  and  the  deviations  of  the  g's  for  any  other 
choice  of  coordinates  may  be  looked  upon  as  due  to  the  dis- 
tortion of  the  space-time  extension  or  to  the  gravitational 
field. 

There  is  a  general  limitation  imposed  on  the  g's  not  by 
mathematics  but  by  nature;  and  this  limitation  is  expressible 
by  means  of  differential  equations  satisfied  by  the  g's  and  thus 
exhibiting  the  law  of  gravitation;  these  differential  equations 
we  shall  find  later. 

Now  the  g's  vary  with  the  system  of  coordinates  employed. 
But  the  essential  law  is  independent  of  the  system  of  coor- 
dinates. If  new  coordinates  are  chosen  we  ge/  new  values 
of  the  g's  through  use  of  the  transformed  form  of  (7)  and  the 
hypothesis  of  the  invariance  of  ds.  The  differential  equations 
between  the  new  g's  and  their  coordinate  system  must  be 
the  same  as  those  between  the  old  g's  and  their  coordinate 
system.  In  other  words,  the  differential  equations  expressing 
the  law  of  gravitation  must  be  covariant  under  the  general 
transformations  of  axes  which  we  have  defined.  This  fact  will 
furnish  us  with  a  valuable  guide  in  the  development  of  the  new 
laws. 

Moreover,  if  we  have  equations  expressing  some  physical 
law  in  the  usual  coordinates  and  we  are  able  to  recognize  these 
equations  as  the  degenerate  form  of  equations  covariant  under 
our  general  group  of  transformations  we  may  hope  to  be  able 
to  rewrite  the  known  equations  in  a  more  far-reaching  form, 


84  THE  THEORY  OF  RELATIVITY. 

so  that  they  shall  then  afford  the  extended  law  on  the  basis  of 
the  theory  of  relativity.  It  is  by  this  method  of  approach 
indeed  that  we  shall  obtain  the  differential  equations  which 
characterize  the  gravitational  field. 

§  41.  THE  THEORY  OF  TENSORS. 

In  order  to  proceed  with  our  problem  of  finding  the  differen- 
tial equations  of  the  general  gravitational  field  we  shall  need 
the  properties  of  certain  fundamental  mathematical  symbols; 
these  we  treat  in  this  and  the  two  following  sections  in  so  far 
as  they  are  needed  for  our  present  purposes. 

We  shall  be  concerned  particularly  with  differential  equa- 
tions which  are  covariant  under  the  group  of  continuous  trans- 
formations, 

(9)  */=/*(*!,  *2,  *3,  *4),      *=  I,  2,  3,  4. 

This  is  due  to  the  fact  that  in  the  general  theory  of  relativity 
the  laws  of  nature  are  to  be  expressed  in  a  form  which  is  co- 
variant under  the  transformations  (9).  Here  we  assume  that 
the  f  unctions /f  are  of  such  sort  that  the  Jacobian 


of  the  transformation  is  different  from  zero  throughout  the  entire 
range  of  values  in  consideration.  That  these  transformations 
form  a  group  follows  from  the  existence  of  a  unique  inverse  of 
the  transformation  (due  to  the  non-vanishing  of  the  Jacobian) 
and  the  fact  that  the  product  of  two  transformations  of  the 
set  also  belongs  to  the  set. 
Now  in  view  of  (9)  we  have 


where  0  is  a  function  of  xi,  X2,  xs,  x±  and  Sj  denotes  the  sum 
as  toy  for  y  =  i,  2,  3,  4.  (We  shall  employ  similarly  the  symbols 
2M,,  Za/3T,  etc.,  to  denote  the  sum  of  the  elements  formed 


THE  GENERALIZED  THEORY  OF  RELATIVITY.  85 

from  the  term  following  the  symbol  by  giving  independently  to 
the  suffixes  /*,  v  or  a,  0,  7,  etc.,  the  values  i,  2,  3,  4.)  We  may 
look  upon  these  equations  as  transforming  the  vectors  (dxi, 
dx-2,  dxs,  dx±)  and  (9</>/9#i,  .  .  .  ,  3^/8^4)  into  the  vectors 
(dxi,dx2,dxz,dxi)  and  (90/d*i',  .  .  .  ,  8<£/8#4')>  respectively. 
If  two  vectors  A*  and  A^  are  transformed  through  (9)  by 
the  first  and  second  of  these  laws,  respectively,  so  that  we  have 


we  shall  say  that  A"  is  a  contravariant  vector  and  that  A^  is  a 
covariant  vector.  Moreover,  we  shall  uniformly  mean  by  the 
symbols  A  ",  B»,  .  .  .  contravariant  vectors;  and  by  the  symbols 
Ap,  Bp,  .  .  .  covariant  vectors. 

If  IJL  and  v  each  ranges  over  the  set  i,  2,  3,  4,  the  symbol  A^ 
will  denote  a  quantity  having  sixteen  components.  Similarly 
the  symbols  A^  and  A'^  will  each  denote  a  quantity  having 
sixty-four  components.  We  may  look  upon  such  quantities  as 
a  sort  of  generalized  vectors.  Since  the  present  theory  is  con- 
cerned especially  with  those  for  whose  components  the  trans- 
formation equations  are  linear  and  homogeneous  it  is  found 
convenient  to  apply  a  particular  name  to  such  vectors;  and 
they  are  called  tensors.  If  then  a  law  of  nature  is  so  formulated 
as  to  be  expressed  through  the  vanishing  of  all  the  components 
of  a  tensor,  it  will  be  covariant  under  the  transformations  (9). 
Tensors  are  therefore  of  central  inportance  for  the  theory  of 
relativity. 

By  an  extension  of  the  terminology  employed  in  connection 
with  (10)  we  may  speak  of  covariant,  contravariant  and  mixed 
tensors;  those  for  two  indices  (or  those  of  rank  two)  obey  by 
definition  the  transformation  laws: 

^./y         ^<V 

(i  i)  A  '„„  =  2aT  —  -,  —^A  ar       (covariant  tensor)  , 

0%n    OXV 

(12)  A'"  =  2ffT  |^  ^-(contravariant  tensor), 


86  THE  THEORY  OF  RELATIVITY. 

(13)  A';  =  2.,^,^    A;        (mixed  tensor). 

C#M     C%T 

It  will  be  observed  that  the  notation  for  each  type  of  tensor 
is  so  chosen  as  to  indicate  the  character  of  the  tensor.  Tensors 
of  the  third  and  higher  rank  are  so  defined  that  analogous  laws 
of  transformation  obtain;  thus  for  co  variant  tensors  of  the 
third  rank  we  have  the  law 

A  t        _  y        ^XP     CX<r    3^L  A 

r 


Vectors  such  as  those  involved  in  (10)  may  be  called  tensors 
of  rank  unity.  A  scalar  (invariant)  may  be  called  a  tensor 
of  rank  zero  and  classed  as  either  covariant  or  contravariant. 

If  we  now  introduce  a  third  set  #x"  of  coordinates  as  functions 
of  the  set  x\  and  if  A^f  is  the  transformed  vector  of  A^  when 
transformed  by  this  substitution,  we  have 


_ 

->  f 

But 


From  this  it  follows  that  the  final  result  obtained  by  apply- 
ing the  two  transformations  successively  to  Aff  is  the  same  as 
that  obtained  by  applying  at  once  the  product  of  the  two 
transformations.  It  is  not  difficult  to  see  that  this  transitive 
property  is  possessed  by  the  tensors  of  each  of  the  several  classes. 

It  is  evident  that  the  sum  of  any  two  tensors  of  the  same 
given  character  is  also  a  tensor  of  that  character. 

It  may  also  be  proved  that  the  product  of  two  tensors  is 
a  tensor  and  that  its  character  is  the  sum  of  the  characters 
of  the  two  component  tensors.  Let  us  prove  this  for  the  single 
case  of  a  product  of  the  form  A^B^.  From  the  relations  of 
transformation 


THE  GENERALIZED  THEORY  OF  RELATIVITY.  87 

we  have 


(  Af      D'P^_V  a  0  y  p  (A       T>    5\ 

(A  ^  )  -  ^^  —,  —  ,  —  (4.0,  ). 

Hence  the  law  of  transformation  is  that  of  a  tensor  of  the 
fourth  rank  having  the  character  denoted  by  the  symbol  CJU. 

In  particular,  the  product  of  two  vectors  is  a  tensor  of  the 
second  rank;  there  are  also  tensors  of  the  second  rank  which 
are  not  products  of  two  vectors. 

In  the  two  preceding  paragraphs  we  used  the  term  product 
of  two  tensors  to  denote  the  tensor  whose  component  elements 
are  all  the  elements  formed  by  multiplying  an  element  of  one 
tensor  by  an  element  of  another  tensor.  This  may  be  called 
their  outer  product.  We  need  also  the  notion  of  inner  product 
of  two  vectors,  say  of  A^  and  J3";  and  this  is  defined  to  be  the 
quantity  S^^^j  that  is,  the  sum  of  products  of  correspond- 
ing elements. 

From  a  mixed  tensor  such  as  A^we  can  form  a  contracted 
tensor  ^ffA^0  which  we  denote  by  B^.  Let  us  prove  that 
the  last  quantity  is  indeed  a  tensor.  We  have 


But 

o  if 


'°  fix,'  3*5       8*5         i  if  7=5. 
Hence, 


r    1  5 


Therefore, 


showing  that  B^  is  a  tensor  of  the  second  rank  of  the  character 
already  anticipated  in  the  notation.  This  process  of  con- 
traction is  applied  to  remove  from  the  symbol  any  two  indices. 
of  different  character  but  never  two  of  the  same  character. 


88  THE  THEORY  OF  RELATIVITY. 

To  prove  that  a  given  quantity  is  a  tensor  of  given  character 
we  may  merely  verify,  as  in  the  preceding  paragraph,  that  its 
equations  of  transformation  are  those  by  which  the  tensor 
character  is  defined.  But  the  result  may  often  be  more  readily 
obtained  by  the  use  of  the  following  theorem  :  //  the  inner  product 
of  a  given  quantity  by  every  covariant  (or  by  every  contravariant) 
vector  is  a  tensor  then  the  given  quantity  is  itself  a  tensor. 

The  general  method  of  argument  will  be  seen  from  the 
following  special  case:  Suppose  that  ^VAIJLVBV  is  a  covariant 
vector  for  every  choice  of  the  contravariant  vector  Bv.  Then 
we  have 


the   last    relation    coming    from    the    inverse    transformation. 
Therefore, 


Since  Bfv  is  arbitrary  it  follows  that  the  parentheses  quantity 
must  have  the  value  zero  for  every  value  of  /*  and  vf  showing 
that  ApV  is  a  covariant  tensor  of  rank  two.  A  similar  proof 
can  evidently  be  made  for  tensors  of  any  given  character. 

Now  in  accordance  with  the  theory  of  relativity  the  general 
expression  for  ds2, 

ds2  =  Z^g^dx^dx,,,      g^  =  gvl» 

is  to  be  invariant  under  our  transformations.  Here  dx^  plays 
the  role  of  an  arbitrary  contravariant  vector,  whence  it  follows 
by  use  of  the  preceding  theorem  that  ^vg^vdxv  is  a  covariant 
tensor  of  the  first  rank.  Repeating  the  argument  with  respect 
to  this  tensor  we  see  that  g^v  is  a  covariant  tensor  of  the  second 
rank.  We  call  it  the  fundamental  covariant  tensor  on  account 
of  its  central  position  in  the  theory  of  gravitation. 

If  in  the  determinant  gH&wl  we  ta^e  the  cof  actor  of  the 
element  g^v  and  divide  this  by  the   determinant  g  we  obtain 


THE  GENERALIZED  THEORY  OF  RELATIVITY.  89 

certain  quantities  gM"(gM"=g''/0;  these  define  a  contra  variant 
tensor,  as  we  shall  now  show.  We  call  it  the  fundamental 
contravariant  tensor.  From  the  theory  of  determinants  we  have 
^ffgnffgvff  =  g^y  where  g/  is  i  or  o  according  as  //  and  v  are  equal  or 
unequal.  Since  *L9gjA*  =  Av  for  every  contravariant  vector  A", 
it  follows  that  g/  is  a  mixed  tensor  of  the  character  anticipated 
by  the  notation.  We  call  it  the  fundamental  mixed  tensor. 
We  now  have  readily 


ds2  =  Z^g^ 
Introducing  the  notation  d£ff  =  2ltgliadxll,  we  have 


Since  d%ff  is  an  arbitrary  covariant  vector  it  follows  from  the 
last  relation  and  the  fact  that  gffr  =  gTff  that  gffT  is  a  contravariant 
tensor  of  the  character  indicated. 

With  any  covariant  tensor  A^  we  may  have  the  following 
two  associated  tensors  and  scalar  of  character  indicated  by  the 
notation  : 

(14)  A;  =  ^agvaA^      A»v  =  Vag"*Aav,      A  =  2,As. 

If  g'  ^  are  the  quantities  into  which  the  g^  transform 
through  (9)  and  if  g  and  g'  denote  the  determinants  |  g^  \  and 
|  g'^  |  while  /  is  the  Jacobian  of  the  transformation  we  have 
readily  from  the  theory  of  determinants  that 


If  dr  and  drf  are  the  elements  of  four-dimensional  volume  in 
our  space-  time  extension  we  have  drf  =  Jdr,  so  that  from  the 
foregoing  relation  it  follows  that 


Now  if  we  employ  our  hypothesis  that  in  infinitely  small 
regions  the  special  relativity  theory  is  valid  we  see  that  coor- 
dinates may  be  chosen  so  that  ds2  has  the  form  given  in  (5),  the 


90  THE  THEORY  OF  RELATIVITY. 

corresponding  determinant  g  having  the  value  —  i.  If  dro 
denotes  the  volume  element  in  this  system,  the  "natural"  volume 
element,  we  have  dro  =  V— g  dr.  We  see  that  g  cannot  vanish 
since  then  we  should  have  for  a  finite  volume  element  dr  an 
infinitesimal  volume  element  dro.  We  shall  assume  that  g  has 
always  a  finite  negative  value,  this  assumption  being  in  agree- 
ment with  the  special  theory  of  relativity. 

Since  —  g  is  always  positive  and  finite  there  must  exist  a 
set  of  axes  for  which  it  has  the  value  i.  Expressed  in  terms  of 
such  a  set  of  axes  the  laws  of  nature  will  have  a  particularly 
simple  form.  One  should  first  derive  them  in  their  general 
covariant  form  and  afterwards  simplify  them  by  the  special 
choice  of  axes  indicated,  the  simplification  being  effected  merely 
as  a  matter  of  convenience. 

§  42.  COVARIANT  DIFFERENTIATION. 

The  writing  of  certain  important  covariant  differential 
expressions  is  greatly  facilitated  by  the  use  of  Christoffel's 
3-index  symbols,  namely, 


[w, 


•tH.cc    i      (->6  vat          U£ 
-  -j 

These  symbols  satisfy  the  relations 

By  [fjiVj  X]'  and  \nv,  X}'  we  denote  the  quantities  obtained 
from  \JJLV,  X]  and  {pv,  X}  respectively  on  replacing  #M  by  xj 
and  g^  by  g'^. 

If  in  the  relation 


one  differentiates  with  respect  to  #x',  «M'»  #/>    anc^  subtracts 
member  by  member  the  first  resulting  relation  from  the  sum 


THE  GENERALIZED  THEORY  OF  RELATIVITY.  91 

member  by  member  of  the  last  two,  one  has  a  result  which  is 
readily  put  in  the  form 


r      xl,     v  a        0   ,v         a       0       yr        -, 

fry,  \\  =2a0ga0       ,       ,  —  -,  +  ^a/37-—  ,  —,—,100,  y\. 

d#M  ox"  ox\  oxn  oxv  ox\ 

Multiplying  by  g'Xp8#e/8#p',  summing  as  to  X  and  p,  and 
simplifying  by  use  of  relations  (12),  (15),  and  properties  of  the 
g's  given  in  the  paragraph  ending  with  (14),  we  have 

/  t\  v  f         )/3#«          8  xe     .  v    3#a  3^0  ,         , 


On  differentiating  a  scalar  quantity  one  obtains  a  covariant 
tensor  of  rank  unity;  but  on  differentiating  a  tensor  of  rank 
greater  than  zero  one  obtains  a  quantity  which  is  not  necessarily 
a  tensor.  It  is  therefore  desirable  to  define  a  process  generalizing 
that  of  differentiation  and  of  such  sort  as  to  lead  always  from  a 
given  tensor  to  a  new  tensor.  Such  a  process  we  now  define. 

If  we  differentiate  with  respect  to  xj  both  members  of  the 
second  equation  in  (10)  and  in  the  result  replace  the  second 
derivatives  by  their  values  taken  from  (16)  we  have 


Simplifying  the  second  term  of  the  first  member  by  means 
of  the  second  relation  in  (10)  and  introducing  the  symbol  A^ 
with  the  meaning 

(18)  A»v  =  d^-2p\^p\ApJ 

ox» 

we  see  from  (17)  that  A^  is  transformed  in  accordance  with 
relations  (n)  so  that  A^  is  a  tensor  of  the  character  indicated 
by  its  symbol.  This  tensor  A^  is  called  the  covariant  derivative 
oiA^. 

In  a  somewhat  similar  manner  one  can  obtain  formulae  for 


92  THE  THEORY  OF  RELATIVITY. 

defining  the  covariant  derivatives  of  contravariant  and  mixed 
tensors,  namely, 

(19)  4,» 

(20)  X;* 


In  a  similar  way  one  introduces  also  the  covariant  derivative 
v  of  A^  by  means  of  the  definition 


(22) 


In  each  case  it  may  be  shown  that  the  covariant  derivative 
has  the  character  indicated  by  the  notation  adopted  for  it. 

Whenever  the  Christoffel  symbols  vanish  the  covariant 
derivatives  reduce  to  ordinary  derivatives;  they  so  reduce,  in 
particular,  when  the  g's  have  Galilean  or  any  other  constant 
values. 

§  43.  THE  RIEMANN-CHRISTOFFEL  TENSOR. 

Let  Ap  be  any  covariant  tensor  of  rank  unity.  Form  its 
covariant  derivative  A^  in  accordance  with  equation  (18). 
Form  the  covariant  derivative  A^  of  A^v  in  accordance  with 
equation  (22),  employing  the  form  of  A^v  given  in  (18).  Thus 
we  have 

_   a2^   _^  .    v      \dAjL_z  I    a    el— -Z  \vo    c*9^" 


\{ev,  p}Ap-2pAp-  {»„,  p}. 


In  view  of  the  definitions  of  the  Christoffel  symbols  it  is 
seen  that  the  first  five  terms  in  the  second  member  are  unaltered 


THE  GENERALIZED  THEORY  OF  RELATIVITY.  93 

by  an  interchange  of  v  and  a.  Hence  the  tensor  A^va—  A^ffv  has 
the  value  ZpB^Ap  where 

(23)        5Jw=2€(/i(r,  c}{ey,  pj-Se(/i",  die*,  pi 

+_9_{^pj_Aj          I 
dx,  Qxff 

Since  ^4P  is  arbitrary  and  2pBp^ffAp  has  the  tensor  character 
BHW  it  follows  that  B^V<T  has  the  tensor  character  anticipated 
by  the  notation  employed.  It  is  called  the  Riemann-Chr  is  to/el 
tensor.  It  will  be  observed  that  this  tensor  depends  upon 
nothing  but  the  fundamental  tensor  &,„. 

Now  if  we  have  a  physical  situation  in  which  it  is  possible 
to  choose  the  coordinate  system  so  that  the  coefficients  g^  shall 
be  constants,  then  for  this  system  the  Bpva.  must  vanish.  From 
this  and  its  tensor  character  it  follows  that  it  must  also  vanish 
however  the  coordinate  system  is  transformed  in  accordance 
with  equations  (9).  The  vanishing  of  this  tensor  is  then  a  neces- 
sary condition  that  it  shall  be  possible  to  choose  the  system  of 
reference  in  such  wise  that  the  g^  shall  have  their  constant 
Galilean  values.  It  may  be  shown  (though  we  do  not  here  give 
the  proof)  that  the  condition  is  also  sufficient. 

From  this  it  follows  that  in  our  problem  the  vanishing 
of  the  Riemann-Christoffel  symbol  corresponds  to  the  possibility 
of  choice  of  coordinates  such  that  the  special  theory  of  relativity 
shall  be  valid  in  a  finite  region. 

If  we  employ  the  notation 


a2  log  V- 


we  have  without  difficulty  the  relation 

(24J  Bfil,  =  'ZTBTtJll,T=R 


In  order  to  effect  the  simplification  in  the  result  stated  here 
one  needs  the  following  value  of  2p{^p,  pj  : 


•94  THE  THEORY  OF  RELATIVITY. 

[P«  ,    dg(*_  \ 


the  second  last  member  being  obtained  by  use  of  the  fact  that 
g^g  is  the  cofactor  of  gp€  in  the  determinant  g. 

It  may  readily  be  shown  that  R»v  and  SMJ,  have  the  tensor 
character  indicated  by  the  notation. 

Now  we  have  seen  that  the  coordinate  axes  may  be  chosen 
so  that  V  — g  has  the  value  i.  Under  such  choice  several  of 
the  preceding  formulae  become  simpler.  This  is  particularly 
true  of  the  expression  for  B^  since  S^  then  has  the  value  zero. 
This  simplification  is  of  considerable  importance  in  the  theory 
of  gravitation  on  account  of  the  fundamental  role  in  this  theory 
of  the  tensor  B^. 

§  44.  EINSTEIN'S  LAW  OF  GRAVITATION. 

We  have  seen  (in  §  40)  that  the  values  of  the  g's  which 
obtain  in  the  absence  of  a  gravitational  field  and  for  a  suitable 
choice  of  reference  system  are  the  constant  Galilean  values. 
In  §  43  we  saw  that  the  vanishing  of  the  Riemann-ChristofTel 
tensor  B^vff  is  a  necessary  and  sufficient  condition  that  it  shall 
be  possible  to  choose  the  system  of  reference  in  such  wise  that 
the  g^  shall  have  the  constant  Galilean  values.  Hence  a  neces- 
sary and  sufficient  condition  for  the  absence  of  a  permanent 
gravitational  field  is  the  following: 


(25)  S.lAKT,  e!fe^  p\-^^v,  e}\ev,  p\ 


-—  {/ii>,  P(=O. 


Of  the  96  relations  obtained  from  the  six  effectively  distinct 
combinations  of  a  and  v  and  the  16  combinations  of  ju  and  p, 


THE  GENERALIZED  THEORY  OF  RELATIVITY.  95 

only  20  are  independent.  If  we  write  (prvv)  for  S^pB;,,,  so 
that  Bp^<T  =  ^Xp(fj.\av'),  it  is  seen  that  equation  (25)  is  equiv- 
alent to  the  equation  (JJLT<TV)=O.  The  reduction  to  20  inde- 
pendent relations  is  then  effected  through  use  of  the  identities 


(IJLT<TV)  +  (fJLavr)  +  (^vrd)  =  O. 

Now  the  general  law  of  gravitation  must  contain  as  a 
special  case  that  expressed  by  the  vanishing  of  the  Riemann- 
Christoffel  symbol  and  must  itself  be  expressed  in  the  form 
of  differential  equations  satisfied  by  the  g's.  One  of  the  simplest 
conditions  meeting  these  requirements  is  that  expressed  by  the 
vanishing  of  the  tensor  B^v  defined  in  equations  (24).  This 
yields  the  equation 


(26)  £M,s-S        -W,  pl  +  Sj/ip,  e\{ve,  p] 

(jOOp 

+  ^T  log  V~S-^^",  «l£  log  V=^=0. 

O^n  O*v  O%e 

Since  B^  —  B^  there  are  in  (26)  only  ten  different  equations; 
and  among  these  there  are  four  identical  relations,  so  that 
only  six  of  the  equations  are  independent.  These  equations 
are  taken  by  Einstein  in  the  general  theory  of  relativity  as  the 
mathematical  expression  of  the  law  of  gravitation  in  the  absence 
of  matter  and  the  electromagnetic  field.  [The  reader  should 
observe  the  marked  simplification  of  the  equations  when  the 
reference  system  is  so  chosen  that  V—  g  =  i.] 

Einstein  insists  that  there  is  a  minimum  of  arbitrariness 
connected  with  this  choice  of  equations.  For  B^  is  the  only 
tensor  of  second  rank  which  is  formed  from  the  g^v  and  their 
first  and  second  derivatives  and  is  linear  in  the  second  deriva- 
tives. Moreover,  no  tensor  of  lower  rank  can  be  built  up  out 
of  the  components  of  B£v<r  by  allowable  processes.  One  who 
counts  up  all  the  terms  represented  by  the  various  symbols 
in  (26)  will  probably  admit  that  we  should  first  find  out  whether 
the  suggested  law  of  gravitation  is  valid  before  trying  a  more 


96  THE  THEORY  OF  RELATIVITY. 

complicated  one.  But  it  must  be  remembered  that  the  general 
theory,  apart  from  facts  of  observation,  does  not  lead  necessarily 
to  this  particular  law. 

In  the  Newtonian  theory  of  attraction  it  is  the  Laplace 
equation  A2<£  =  o  which  corresponds  to  our  equation  (26).  The 
covariant  equation  corresponding  to  Poisson's  equation  A20  = 
— 4?rp  has  also  been  treated  by  several  writers  (including 
Einstein) ;  but  we  shall  not  develop  it  here. 

§  45.  THE  MOTION  OF  A  PARTICLE. 

Denote  by  Aff  the  contravariant  vector  dx*/ds.  Multiplying 
Aa  by  the  covariant  derivative  Aa°  of  Aff, 


and  summing  as  to  a,  we  have 


From  this  equation  it  follows  that  the  expression  in  the 
second  member  is  a  contravariant  vector.  Let  us  consider  the 
equation  obtained  by  setting  it  equal  to  zero,  namely, 

/    \  d2xff 


On  account  of  the  covariant  character  of  this  equation  it  is 
satisfied  or  not  independently  of  the  choice  of  coordinates. 
In  the  case  of  the  special  theory  of  relativity  the  Christoffel 
symbols  have  the  value  zero  and  the  equations  reduce  to 
d2xa/ds2  =  o;  and  these  are  equations  of  a  straight  line.  In 
this  special  case,  then,  equations  (27)  give  the  path  of  a  moving 
particle  in  the  absence  of  a  permanent  gravitational  field. 
Accordingly,  in  view  of  our  principle  of  equivalence,  we  assume 
that  equations  (27)  are  the  equations  of  motion  of  a  particle 
referred  to  any  axes,  even  though  there  is  a  permanent  gravita- 
tional field.  We  are  justified  in  this  in  view  of  the  fact  that 


THE  GENERALIZED  THEORY  OF  RELATIVITY.  97 

the  equations  contain  no  derivative  of  the  g^  of  order  higher 
than  the  first. 

Let  us  now  see  in  what  sense  the  Newtonian  theory  of  the 
motion  of  a  particle  is  a  sort  of  first  approximation  to  the  theory 
based  on  equations  (27).  In  the  case  of  the  special  theory  of 
relativity  the  components  dx\/dsy  dxz/ds.  dxs/ds  of  the  velocity 
v  can  have  arbitrary  values.  If  the  velocity  of  light  is  taken 
to  be  unity  and  v  is  a  very  small  quantity  then  its  components 
are  small  while  dx±/ds  is  equal  to  unity  except  for  quantities 
of  the  second  order.  Moreover,  in  the  limiting  case  of  the 
special  theory  of  relativity  the  quantities  g^  have  the  value 
zero  when  p^v,  while  gn=-i,  #22=-i,  #33=-!,  #44  =  i. 
From  this  point  of  view  the  quantities  {a/3,  a\  are  of  order  at 
least  as  high  as  the  first.  From  these  results  Einstein  con- 
cludes that  in  equation  (27)  the  desired  approximation  is  to  be 
attained  by  considering  only  that  gMJ,  for  which  ^  =  4  =  ^  and  is 
then  led  to  the  approximate  equations 


d2xff 
-l44,4l,       -^  =  144,*},       <r  =  i,  2,3, 

by  taking  the  approximate  relation  ds  =  dt. 

If  we  assume  further  that  the  gravitational  field  is  quasi- 
static  in  the  sense  that  the  acceleration  due  to  the  gravitational 
field  is  very  small  compared  with  the  velocity  of  light  so  that 
derivatives  with  respect  to  the  time  may  be  neglected  in  com- 
parison with  those  with  respect  to  the  space-coordinates,  we 
have  approximately 


_  _ 

~ 


These  are  the  equations  of  motion  of  a  material  particle  in 
the  Newtonian  theory  in  which  #44/2  plays  the  role  of  the 
gravitational  potential.  It  is  noteworthy  that  these  approxi- 
mate equations  depend  on  the  single  component  #44  of  the 
fundamental  tensor. 

Going  back  to  the  general  point  of  view  of  equations  (27) 


98  THE  THEORY  OF  RELATIVITY. 

and  accepting  the  guidance  of  our  current  ideas  (just  seen  to 
be  approximately  valid),  let  us  consider  the  case  of  a  particle 
at  rest  at  the  origin  of  coordinates  in  our  space-  time  extension. 
We  depart  somewhat  from  the  strict  standpoint  of  the  general 
theory  of  relativity  and  choose 

(28)  %i=r,      x2  =  6,      #3  =  0,      %±  =  t 

as  our  coordinates,  treating  them  as  the  usual  polar  coordinates. 
Then  ds2  may  be  assumed  to  have  the  form 

ds2=-e*dr2-ef'(r2dd2+r2sm2  dd<f>2)+evdt2, 

where  X,  M,  v  are  functions  of  r  only. 

One  may  justify  the  omission  of  the  product  terms  drdQ 
and  drd$  and  d6d<t>  by  the  symmetry  of  the  polar  coordinates, 
and  the  omission  of  drdt  and  dBdt  and  d$dt  by  the  symmetry  of 
a  static  field  with  respect  to  past  and  future  time. 

If  we  write  r2el*  =  r'2  and  absorb  into  the  X  the  resulting 
change  in  dr2  and  then  write  r  for  the  new  /  we  have  for  ds2  the 
expression 

(29)  ds2=  -e*dr2-r2dd2-r2  sin2  ed<t>2+e"dt2. 
From  this  we  have  gMJ,  =  o  when  ^^v  and 

(3°)          gn=e-*,    g22=-r2,    #33=  -r2  sin2  6,    g±±  =  ev. 

Here  X  and  v  are  functions  of  r. 

The  determinant  g  reduces  to  its  main  diagonal  and  we  have 


The  three-index  symbol  {or,  «}  has  the  value 


If  (7,  T,  p  are  three    distinct  numbers  we  have    {or,  p}=o 
while 


THE  GENERALIZED  THEORY  OF  RELATIVITY.  99 

i-^ 


By  a  straightforward  computation  one  may  now  evaluate 
the  Christoffel  symbols  involved  in  (26)  and  so  obtain  the 
explicit  form  of  those  equations  for  the  special  case  now  in 
consideration.  It  turns  out  that  BfJLt,  =  o  is  identically  satisfied 
when  n^v.  From  the  four  equations  in  which  /z  =  i>  we  have 
by  the  indicated  direct  computation  the  relations 


(32) 


where  the  primes  denote  differentiation  with  respect  to  r. 

Combining  the  first  and  last  equations  we  see  that  X'  =  —  v  . 
Then,  since  X  and  v  must  tend  to  zero  as  r  tends  to  infinity  so 
that  gn  and  #44  shall  have  at  infinity  the  Galilean  values  —  i 
and  +  1  ,  respectively,  it  follows  that  X  =  —  v.  Then  the  second 
and  third  equations  in  (32)  both  reduce  to  the  equation 


Solving  this  we  have  e"  =  i  —  2m/r,  where  2m  is  a  constant 
of  integration.  This  solution  also  satisfies  the  first  and  fourth 
equations  in  (32).  Substituting  in  (29)  the  derived  values  for 
e*  and  ev  we  have 

(33)        ds2  =  -  (i  -  ™}    *dr2  -  r2d02  -  r2  sin2  6d<t>2 


Substituting  into  (27)  the  values  of  the  Christoffel  symbols  in 
the  same  special  forms  as  we  have  just  employed  in  (26)  we 


100  THE  THEORY  OF  RELATIVITY. 

may  obtain  the  explicit  forms  of  equations  (27)  for  our  present 
problem.     Equation  (27)  for  <r  =  2  thus  becomes 


d26  J 

(34)  —  -cos0sin0( 

ds2  \ 


.   2drdd 

—  -cos0sin0-    1  +-  —  —=o. 
ds/       r  ds  ds 


If  we  choose  coordinates  so  that  the  particle  moves  initially  in 
the  plane  0=?r/2  we  have  initially  dB/ds  =  o  and  cos  0  =  o,  so 
that  d2d/ds2  =  o;  whence  it  follows  that  the  particle  continues 
to  move  in  this  plane.  With  such  a  choice  of  coordinates 
equations  (27)  for  o-  =  i,  3,  4  take  the  forms 

d2r  ,  id\/dr\2          Jd<t>\2  .  ±  ,     ,dv/dt\2 

-  )  ^       drdS     =°' 


= 
ds2     rds  ds°'       ds2     dsds~ 


From  equations  (36)  we  have 
J<  dt 


where  h  and  7  are  constants  of  integration  and  where  in  the  last 
member  we  have  replaced  e~v  by  its  value  obtained  in  the 
paragraph  ending  with  equation  (33).  Again,  if  we  replace  X 
and  v  in  (35)  by  their  values  previously  derived  we  have 


Since  dd  =  o  and  6  =  ir/2  we  now  have  from  (33)  the  relation 

2  2  2 


From  this  relation  and  (37)  we  have 

/dr\*.    2/d4>\2  ,  2m  ,       h2 

(ds)    +r(d-S)    =T2-I+-  +  2W-. 


THE  GENERALIZED  THEORY  OF  RELATIVITY.  101 

If  in  equation  (40)  we  replace  d4>/ds  by  its  value  from  (37) 
and  then  differentiate  with  respect  to  s  we  have  for  d2r/ds2 
the  same  value  as  that  obtained  from  (38)  on  eliminating  the 
first  derivatives  in  (38)  by  aid  of  (37)  and  (39).  Hence,  if  we 
retain  equations  (37)  and  (40)  we  may  omit  equation  (38). 
Then  equations  (37)  and  (40)  are  the  sole  equations  of  motion. 

In  the  corresponding  coordinates  the  Newtonian  equations 
of  elliptic  motion  are 


<••>          aa"  =*- 

To  make  the  first  of  these  correspond  with  (40)  we  must 
regard  ds  as  replacing  dt  and  take  for  y2  the  value  y2  =  i—m/a, 
a  being  the  semimajor  axis  of  the  orbit.  The  term  2mh2/r* 
in  (40)  represents  a  small  additional  effect  not  in  evidence  in 
the  Newtonian  theory.  The  quantity  m,  previously  intro- 
duced as  a  constant  of  integration,  is  now  to  be  identified  as  the 
mass  of  the  attracting  particle  measured  in  gravitational  units. 

§  46.  THREE  CRUCIAL  PHENOMENA. 

If  we  take  one  kilometre  as  the  unit  of  length  and  choose 
the  unit  of  time  so  that  the  velocity  of  light  is  unity,  we  obtain 
from  (41)  the  approximate  value  m  =  1.47  for  the  mass  of  the  sun 
on  supposing  that  the  path  of  motion  of  the  earth  is  circular  so 
that  r  =  a=  i  ^g.io8  and  on  employing  the  value  co  =  6  -64.10  ~13 
of  the  angular  velocity  d^/dt.  Hence  m/r  is  of  the  order 
io~8.  Moreover,  it  follows  from  the  second  equation  in  (41) 
that  h2/r2  is  approximately  equal  to  co2a2  and  is  hence  of  order 
io-  8. 

From  (40)  and  the  first  equation  in  (37)  we  have 


(h  dr  \2  ,  h2  ,  2m  ,       h2 

-5-77)   +-5- =7  —  H h2W— . 

\r2d^/      r2  r  r3 


If  we  transform  this  equation  by  use  of  the  relation  r—i/u 
and  differentiate  both  members  of  the  resulting  equation  with 
respect  to  <£,  we  have 


102  THE  THEORY  OF  RELATIVITY. 

/      x  d2U 

(42) 


Now  h2u2  is  a  quantity  of  the  order  of  io~8;  hence,  we  may 
get  a  roughly  approximate  solution  of  (42)  by  neglecting  the 
term  $mu2.  This  gives 


(43)  M  =  J-I^~e  cos  fa-w)]» 

where  e  and  w  are  constants  of  integration.  In  order  to  get  a 
second  approximation  to  the  solution  we  substitute  the  value 
of  u  in  the  last  term  of  (42)  and  obtain  the  equation 


Through  use  of  the  approximate  value  of  u  in  (43)  and  the 
fact  that  h2u2  is  of  order  io~8,  it  may  be  seen  that  the  third 
and  fourth  terms  in  the  second  member  of  the  last  equation 
cannot  produce  appreciable  effects.  But  the  second  term  is  of 
a  suitable  period  to  produce  an  increasing  effect  by  resonance. 
Retaining  from  the  second  member  only  the  first  and  second 
terms  and  solving  the  equation  so  curtailed  we  have 

/yyt  *  *' 

u  =  —  [i-Ncos  O- 


. 

Writing  8w  for  $m2$/h2  and  neglecting  the  second  and  higher 
powers  of  dw,  we  may  put  the  approximate  value  of  u  in  the 
form 


—ie  cos   <>—w— 


Applying  this  to  the  case  of  a  planet  moving  around  the  sun 
we  find  that  while  the  planet  moves  through  one  revolution 
the  perihelion  advances  by  a  fraction  of  a  revolution  equal  to 


THE  GENERALIZED  THEORY  OF  RELATIVITY.  103 

T  being  the  period  of  the  planet  and  c  being  the  velocity  of 
light  in  customary  units  introduced  into  the  last  member  for 
convenience. 

This  formula  gives  42.9,  8.6,  3.8,  1.35  seconds  for  the 
respective  advances  of  the  perihelion  (per  century)  of  the  four 
inner  planets,  Mercury,  Venus,  Earth,  Mars.  This  is  in  close 
agreement  with  observation.  Thus  the  theory  of  Einstein 
yields  a  very  satisfactory  explanation  of  the  celebrated  large 
discordance  in  the  motion  of  the  perihelion  of  Mercury  which 
has  occupied  the  attention  of  astronomers  since  the  time  of 
Leverrier.  There  is  no  trace  of  forced  agreement  in  connection 
with  this  remarkable  success  of  the  theory. 

A  second  crucial  phenomenon  for  the  theory  is  that  of  the 
deflection  of  a  ray  of  light.  In  the  absence  of  a  gravitational 
field  and  with  the  choice  of  coordinates  which  we  have  been 
employing  the  velocity  of  light  has  the  constant  value  unity. 
Hence 


so  that 

ds2  =-dx2-  dy2  -  dz 

Therefore,  for  the  motion  of  light  in  the  absence  of  a  gravita- 
tional field  we  have  ds  =  o;  then  by  the  principle  of  equivalence 
we  must  also  have  ds  =  o  even  in  a  gravitational  field.  Employ- 
ing this  value  of  ds  and  assuming  that  the  path  of  light  is  in  the 
plane  0  =  7r/2  we  have  from  (33)  the  relation 

/       2m\-1/dr\2^  /  d<t>\2  2m 

(44)  (*--)    \dt)  +(r-Tt)  =I~' 

If  v  is  the  velocity  of  light  in  a  direction  making  an  angle  a 
with  the  radius  vector  we  have 


V2 

whence 


/       2m\    1      9         •  9    }           im 
\i 1     cos2  a  -f-  sin2  a  }•  =  I , 

\        r  I  }  r 

(       2m\  (       2m   .  9     1  ~  * 
v=(i H  i sin2 a  \ 

\        r  I  (         r  J 


104  THE  THEORY  OF  RELATIVITY. 

Since  this  gives  a  velocity  for  light  varying  with  the  direction 
we  alter  our  coordinates  slightly  by  replacing  r  by  r +m,  whence 
we  have  that  r2  is  to  be  replaced  by  a  quantity  approximately 
equal  to 

r2/I_±2A 

With  such  a  value  of  r  equation  (44)  oecomes  approximately 
the  equation 

^Jr\2        /  ,7^\2        /         ^^,\2 


( 


, 

+  '•='  - 


whence  we  obtain  for  v  the  approximate  value 


2m 

, 


the  same  in  all  directions. 

If  we  employ  the  principle  that  the  course  of  a  ray  of  light 
depends  only  on  the  variation  of  velocity,  we  find  that  it  will 
be  the  same  as  in  a  Euclidean  space  filled  with  material  of 
a  refractive  index  /*  given  by  the  relation  /JL=I/V,  and  hence 
approximately  by  the  relation 

.  2m 

M  =  H  --  - 
r 

The  gravitational  field  round  a  particle  thus  acts  as  a  con- 
verging lens. 

From  the  foregoing  value  of  /*  it  may  be  shown  without 
difficulty  that  a  ray  of  light  from  —  oo  to  +00,  which  passes 
at  a  distance  R  from  a  particle  of  mass  m,  will  experience  a 
total  deflection  of  amount  4m/  R.  For  the  sun  wre  have 
^  =  1.47  and  R  =  sun's  radius  =  697,000  km.  Hence  for  a  star 
seen  close  to  the  limb  of  the  sun  we  shall  have  a  deflection  of 
1.74  seconds  of  angular  measure. 

This  prediction  was  tested  by  observations  made  inde- 
pendently at  two  stations  during  the  eclipse  of  the  sun  of  May  29, 
1919;  the  values  for  the  deflection  obtained  at  the  two  stations 


THE  GENERALIZED  THEORY  OF  RELATIVITY.  105 

are  1.61  and  1.98  seconds  of  angular  measure,  results  in  sub- 
stantial agreement  with  the  predicted  value. 

Thus  the  Einstein  law  of  gravitation,  as  expressed  in  equa- 
tion (33),  has  been  checked  for  high  velocities  by  the  deflection 
of  a  ray  of  light  and  for  comparatively  low  velocities  by  the 
motion  of  the  perihelion  of  Mercury  —  two  very  remarkable 
conquests  to  be  made  simultaneously  by  a  single  theory. 

A  third  crucial  phenomenon  is  afforded  by  the  vibration 
of  an  atom  in  a  gravitational  field.  Such  an  atom  is  a  natural 
clock  and  should  therefore  give  an  invariant  measure  of  an 
interval  of  time.  If  the  atom  is  at  rest  in  the  system  of  coor- 
dinates (which  themselves  may  be  in  motion)  we  have  dx  =  dy  = 
dz  =  o,  so  that  ds2  =  g44dl2.  If  we  have  two  similar  atoms  at 
different  parts  of  the  field  where  the  potentials  are  #44  and  #'44, 
respectively,  we  have  from  the  invariance  of  ds  that 


If  t  refers  to  the  photosphere  of  the  sun  where  #44  =  i  —  zm/R, 
R  being  the  sun's  radius,  and  t'  refers  to  a  point  on  the  earth 
where  #'44  is  sensibly  equal  to  unity,  we  have  approximately, 

dt         .  m 

—  7  =  i  -f—  =  i  .000002  12. 

dt  R 

Prom  this  it  follows  that  the  atom  vibrates  more  slowly  on 
the  sun  than  on  the  earth,  and  hence  that  the  lines  of  the 
spectrum  should  be  displaced  toward  the  red.  For  the  part 
of  the  spectrum  usually  observed  this  displacement  amounts 
to  about  .008  tenth-meters  (a  tenth-meter  =  io~10  meters). 

There  is  not  yet  agreement  as  to  whether  the  phenomenon 
thus  predicted  is  actually  existent;  in  fact,  some  doubt  has 
been  felt  as  to  whether  the  argument  is  valid  by  which  this 
prediction  is  supported.  C.  E.  St.  John  (Astrophysical  Journal, 
vol.  46,  p.  249)  has  given  negative  evidence  and  L.  Grebe  and 
A.  Bachem  (Deut.  Phys.  Gesell.,  Verh.,  vol.  21,  p.  454)  have 
given  positive  evidence  for  the  existence  of  the  effect.  Einstein 
(quoted  in  Science  for  March  12,  1920,  p.  270)  seems  to  believe 


106  THE  THEORY  Oi'  RELATIVITY. 

that  the  existence  of  the  phenomenon  has  been  established. 
But,  if  it  should  turn  out  that  the  test  fails,  the  most  appro- 
priate conclusion  would  seem  to  be  that  we  have  insufficient 
knowledge  of  the  conditions  of  atomic  vibrations  rather  than 
that  the  theory  of  relativity  is  thus  discredited. 

§  47.  THE  ELECTROMAGNETIC  EQUATIONS. 

That  Maxwell's  equations  may  be  reduced  to  a  covariant 
form  and  hence  that  all  electromagnetic  phenomena  described 
Ly  them  are  in  agreement  with  the  principle  of  relativity  may 
readily  be  shown  in  the  following  way  (the  exposition  being 
based  on  that  of  Eddington,  I.e.,  pp.  76-77): 

The  electromagnetic  field  is  described  by  a  covariant  vector 
KM  which  in  Galilean  coordinates  has  the  components 

(45)  *-(-*,  -G,  -JJ,*), 

where  F,  G,  H  is  the  vector  potential  and  3>  is  the  scalar  potential 
of  the  ordinary  theory.  If  KP,  is  the  covariant  derivative  of 
KM  we  have  by  (18) 

/  f\  8*M     9K, 

(46)  ~""i-~"-> 


a  covariant  tensor  which  we  denote  by  F^. 

The  electric  and   magnetic  forces  of  the  usual  theory  are 
expressed  in  our  present  notation  by  formulas  like 

/_  \                       v    8*1     3*4              9*2     8*3 
U-7 )  A= — ,       a  = . 

8*4     dxi  8*3     8*2 

Hence  in  Galilean  coordinates  the  value  of  F^v  is  given  by  the 
array 

0-7  0       -X 

7  O  —a        —Y 

-|8  a  o        -Z 

X          Y  Z  o 

where  p  varies  through  a  row  and  v  varies  through  a  column. 


THE  GENERALIZED  THEORY  OF  RELATIVITY  107 


The  associated   contra  variant   tensor  F^v  =  ^a^agv^Fa&  is  given 
similarly  by  the  array 

0-7  |8            X 

7              O  —a               Y 

-13            a  o              Z 

-X      -Y  -Z            o 

In   the   ordinary   theory   the  Maxwell   equations   may   be 
written  in  the  form 

/  o\      8^     8 •*           (jet      u-**-      u£  op       9-*      9-^*-          87 

9y    32       a/ '   82     so;  a^ '    8^    ay       a/ ' 


<49)     9l_^=9?+«,    ^-^  =  91+, 
3y    9z     9<  92     9*     9< 


^          if+g+§F- 

/    \  8^  i^  8p  i^  87  _ 

W1*1/  ^^,          '         '-v,.         '         ^,_  ' 


where  the  velocity  of  light  is  taken  to  be  unity  and  the  Heaviside- 
Lorentz  unit  of  charge  is  chosen  so  that  the  factor  4?r  is  absent. 
The  electric  current  M,  v,  w  and  density  p  form  a  contra- 
variant  vector  /M,  since 


,  fdx  dv  dz    dt\ 

J*m(u,V,W,  p)  =  2e(—  ,  -j-,  -,  —  1 

\a5    c?5   ds   ^57 
per  unit  volume.     Equations  (49)  and  (50)  yield  the  relations 


while  equations  (48)  and  (51)  may  be  written 

dFtw.dFvvi  dFffn  _ 
-  —  o. 

8#<r       8*M      dxv 

From  the  definition  of  F^  it  is  seen  that  the  last  equation 
is  satisfied  identically,  so  that  (46)  and  (52)  represent  the  funda- 


108  THE  THEORY  OF  RELATIVITY. 

mental  electromagnetic  equations.  The  former  is  covariant 
and  the  latter  is  rendered  so  on  replacing  the  ordinary  deriva- 
tive by  the  covariant  derivative.  Hence  the  required  co- 
variant equations  take  the  form 

7-1       _  9  KM         9Ky  y    J7M"—    7> 

^*-  —  ~~>       Z^"  ~J  ' 


These  hold  in  the  gravitational  field  because  the  conditions 
for  the  application  of  the  principle  of  equivalence  are  satisfied. 
Here  we  have  shown  that  the  electromagnetic  equations 
are  consistent  with  the  theory  of  relativity,  but  we  have  not 
derived  them  by  means  of  that  theory.  A  more  far-reaching 
treatment  of  the  electromagnetic  problem  has  been  made  by 
Weyl  by  means  of  a  generalized  form  of  the  theory  of  relativity 
(Annalen  der  Physik,  vol.  59  (1919),  p.  101). 

§  48.  SOME  GENERAL  CONSIDERATIONS  RELATING  TO  THE 

THEORY. 

Silberstein  (Philosophical  Magazine,  vol.  36  (1918),  pp. 
94-128)  has  undertaken  to  develop  the  general  theory  of 
relativity  without  the  equivalence  hypothesis.  He  concedes 
that  the  Einstein  theory  has  one  very  strong  point,  namely, 
the  requirement  of  general  covariance  of  all  physical  laws,  that 
actual  phenomenal  contents  should  be  expressed  (or  at  least 
be  expressible)  in  a  way  showing  their  independence  of  the 
particular  language  or  scaffolding  adopted;  but  he  insists  that 
it  has  also  a  weak  point,  namely,  the  equivalence  hypothesis 
which  places  gravitation,  he  believes,  on  an  entirely  exceptional 
and  privileged  footing,  bringing  it  into  intimate  connection 
with  the  fundamental  tensor  which  appears  in  the  line-element 
of  the  world. 

He  proposes  to  retain  the  strong  point  and  to  reject  the 
weak  one  and  thus  to  develop  the  implications  of  the  general 
principle  of  relativity  without  the  equivalence  hypothesis,  in 
fact,  without  privileging  gravitation  at  all.  He  considers  the 
equivalence  hypothesis  to  be  a  vulnerable  point,  independently 


THE  GENERALIZED  THEORY  OF  RELATIVITY.  109 

of  agreement  or  disagreement  with  experimental  facts,  because 
of  its  special  nature  and  of  the  great  number  of  assumptions 
which  it  tacitly  implies. 

It  is  a  matter  of  importance  to  separate  these  two  elements 
and  to  ascertain  to  what  extent  the  results  obtained  in  the 
theory  are  based  on  the  one  or  the  other  of  the  two  parts  of 
the  general  theory.  In  our  treatment  in  the  foregoing  pages 
we  have  followed  the  method  of  Einstein  and  have  not  under- 
taken to  separate  the  two  elements  or  to  distinguish  between 
their  consequences. 

One  matter  not  mentioned  in  our  preceding  treatment 
should  have  at  least  a  word  of  attention.  It  will  be  observed 
that  in  the  theory  as  developed  the  notion  of  gravitational 
force  has  hardly  been  present  at  all  and  that  in  fact  the  proper- 
ties of  the  gravitational  field  are  essentially  geometrical  rather 
than  dynamical  in  character.  Weyl,  in  the  paper  referred  to 
at  the  end  of  §  47,  has  succeeded  in  extending  the  theory  so  as 
to  include  electromagnetic  and  gravitational  forces  in  one 
geometrical  scheme,  thus  extending  the  range  in  which  the 
explanations  may  be  stated  in  purely  geometric  terms. 

When  one  undertakes  to  pursue  to  their  extreme  reach 
the  geometric  conceptions  which  thus  arise  one  is  soon  brought 
to  consider  the  fundamental  character  of  the  four-dimensional 
space-time  extension  by  means  of  which  the  phenomena  are 
thus  interpreted  geometrically.  From  this  point  of  view  one 
seems  forced  to  conclude  that  our  space-time  extension  is  not 
a  flat  space  like  a  plane  or  a  Euclidean  space  of  three  dimensions, 
but  has  an  essential  curvature  involved  in  its  four-dimensional 
continuum  analogous  to  that  of  a  two-dimensional  continuum 
represented  by  a  warped  surface  in  our  usual  space  of  three  dimen- 
sions. 

Some  of  those  who  have  followed  up  this  idea  have  come 
to  the  conclusion  that  our  actual  space-time  manifold  is  finite 
in  extent. 

Several  writers  have  considered  the  possibility  of  founding 
the  entire  theory  of  relativity  on  a  certain  different  basis  from 


110  THE  THEORY  OF  RELATIVITY. 

that  employed  in  this  chapter,  namely,  on  the  principle  of  least 
action.  This  and  the  topic  just  mentioned  previously  are  treated 
in  the  last  two  chapters  of  the  monograph  of  Eddington  already 
referred  to.  For  our  purpose  it  suffices  to  say  that  it  is  possible 
to  formulate  the  principle  of  least  action  naturally  in  such  a 
way  that  our  basic  equations  are  equivalent  to  the  principle. 
From  a  theoretical  point  of  view  there  is  much  to  be  said  in 
favor  of  developing  the  theory  in  this  way;  but  the  purposes 
of  an  elementary  exposition  are  better  served  by  the  plan  of 
treatment  which  we  have  adopted. 

Whatever  may  be  the  final  verdict  as  to  the  validity  of  the 
theory  of  relativity  as  a  whole,  it  seems  practically  certain 
that  it  has  already  made  a  fundamental  and  permanent  con- 
tribution to  astronomy  in  developing  the  modification  of 
Newton's  law  of  gravitation  associated  with  equation  (33),  the 
new  form  of  the  law  now  having  been  checked  experimentally 
in  two  very  different  ways  and  thus  established  on  a  particularly 
secure  experimental  basis.  Two  such  conquests  as  those 
recorded  in  §  46  have  seldom  been  made  so  nearly  simulta- 
neously by  a  single  theory  developed  from  one  point  of  view 
consistently  maintained  throughout. 


INDEX. 


Aberration  of  light,  9. 
Acceleration,  55. 
Action,  26,  59,  no. 
Addition  of  velocities,  46,  73. 
Atomic  vibration,  105. 

Bucherer,  68,  70,  72. 

experiment,  68. 
Bumstead,  51. 

Charge  on  electron,  68-72. 
Christoffel  symbols,  90. 
Conservation  of  electricity,  68. 

energy,  26,  64. 

momentum,  26,  50,  66. 
Conservation,  Postulates  of,  26. 
Consistency  of  postulates,  24. 
Contra  variant  tensor,  85. 
Conventions  in  physical  theory,  7,  34, 

35,  37- 
Covariant  tensor,  85. 

Derived  units,  54. 
Differentiation,  Covariant,  90-92. 
Dimensional  equations,  55. 
Dimensions  of  units,  54. 
Doppler  effect,  18. 

Earth,  Movement  through  ether,  10. 
Eddington,  80,  106,  no. 
Einstein,  44,  77,  95,  97. 
Einstein's  law  of  gravitation,  94-96. 
Electricity,  26,  68. 

Electromagnetic  equations,  106-108. 
theory  of  light,  19. 
Electron,  68-72. 
Emission  theory  of  light,  19. 
Energy,  26. 

and  mass,  49-62. 


Energy,  Conservation  of,  26,  64. 
Equations,  Dimensional,  55. 

of  transformation,  44-48,73.. 
Equivalence,  Principle  of,  79,  108. 
Ether,  9,  10,  14,  61. 
Experimental  verification,  63-72. 
Experiment  of  Bucherer,  68. 

Michelson  and  Morley, 
10. 

Trouton  and  Noble,  13. 

Fizeau,  9. 

Force,  55,  56. 

Foundations  of  physics,  7. 

Fresnel,  9. 

Fundamental  tensors,  88,  89. 

Galilean  axes,  79. 

values  of  g's,  83. 
Gravitation,  62. 

Independence  of  postulates,  24. 

Laws  of  nature  relative  to  observer,  8> 

Length,  27-43. 

Leverrier,  103. 

Lewis,  49. 

Light,  Aberration  of,  9. 

Bending  of  ray  of,  103. 

Mass  of,  62. 

Velocity  of,  18,  47,  58. 
Lines  of  strain,  61. 
Longitudinal  mass,  49. 
Lorentz,  10. 

Mass  and  energy,  49-62. 

velocity,  49. 
Longitudinal,  49,  68. 
Nature  of,  60,  72. 

Ill 


112 


INDEX. 


Mass,  of  electron,  68. 

of  light,  62. 

proportional  to  energy,  58,  59. 

Transverse,  49,  66,  68,  74. 
Matter,  Nature  of,  35,  60. 
Maximum  velocity  of  material  body,  47, 

60. 

Maxwell  equations,  106-108. 
Mechanics,  Systems  of,  9. 
Michelson,  10,  15. 
Michelson-Morley  experiment,  10,  16, 

17- 

Minkowski,  75. 
Mixed  tensor,  86. 
Momentum,  26,  50,  66. 
Morley,'  10,  15. 
Motion  of  a  particle,  96-101. 

Noble,  13,  15. 

Physics,  Foundations  of,  7. 
Perihelion  advance,  101-103. 
Postulate,  C,,  C2,  C3,  26. 

Essential  equivalent  of,  63, 
65- 

L,  22. 

Logical  equivalent  of,  63,  65. 

M,  17. 

R,  17-22,  31. 

V,  22. 

Postulates  of  relativity,  15-26. 

Potentials,  82. 

Principle  of  equivalence,  79,  108. 

least  action,  26,  59,  no. 

Rapidity  of  motion,  74. 
Reference,  Systems  of,  16. 
Relativity  theory  independent  of  ether, 
14. 


Riecke,  24. 

Riemann-Christoffel  tensor,  92-94. 

Robb,  74. 

Silberstein,  108. 
Simultaneity,  38,  40. 
Stewart,  18,  19. 
Strain,  Lines  of,  61. 
Succession  of  events,  38. 
Systems  of  reference,  16. 

Tensors,  84-90. 

Thomson,  19. 

Time  as  a  fourth  dimension,  48,  75. 

Measurement  of,  27-43. 

Nature  of,  37. 
Tolman,  18,  49. 
Top,  Mass  of,  58. 
Transformations,  Equations  of,  44-48, 

73- 

Transformations  in   space  of  four  di- 
mensions, 75,  81. 

Transverse  mass,  49,  66,  68,  74. 

Trouton,  13,  15. 

and  Noble  experiment,  13. 

Units,  Derived,  54. 

Dimensions  of,  54. 
of  length,  32-34. 
of  time,  27-32. 

Velocities,  Addition  of,  46,  73. 
Velocity  and  mass,  49. 

Maximum,  47,  60. 

of  light,  18,  47,  58. 
Verification,  Experimental,  63-72. 

Weyl,  108,  109. 
World-lines,  76-77. 


Wiley  Special  Subject  Catalogues 

For  convenience  a  list  of  the  Wiley  Special  Subject 
Catalogues,  envelope  size,  has  been  printed.  These 
are  arranged  in  groups — each  catalogue  having  a  key 
symbol.  (See  special  Subject  List  Below).  To 
obtain  any  of  these  catalogues,  send  a  postal  using 
the  key  symbols  of  the  Catalogues  desired. 


1— Agriculture.     Animal  Husbandry.    Dairying.     Industrial 
Canning  and  Preserving. 

2 — Architecture.       Building.      Masonry. 

3 — Business  Administration  and  Management.     Law. 

Industrial  Processes:   Canning  and  Preserving;    Oil  and  Gas 
Production;  Paint;  Printing;  Sugar  Manufacture;  Textile. 

CHEMISTRY 
4a  General;  Analytical,  Qualitative  and  Quantitative;  Inorganic; 

Organic. 
4b  Electro-  and  Physical;  Food  and  Water;  Industrial;  Medical' 

and  Pharmaceutical;  Sugar. 

CIVIL  ENGINEERING 

5a  Unclassified  and  Structural  Engineering. 

5b  Materials  and  Mechanics  of  Construction,  including;  Cement 
and  Concrete;  Excavation  and  Earthwork;  Foundations; 
Masonry. 

5c  Railroads;  Surveying. 

5d  Dams;  Hydraulic  Engineering;  Pumping  and  Hydraulics;  Irri- 
gation Engineering;  River  and  Harbor  Engineering;  Water 

Supply. 

(Over) 


CIVIL  ENGINEERING—  Continued 

5e  Highways;  Municipal  Engineering;  Sanitary  Engineering; 
Water  Supply.  Forestry.  Horticulture,  Botany  and 
Landscape  Gardening. 


6 — Design.  Decoration.  Drawing:  General;  Descriptive 
Geometry;  Kinematics;  Mechanical. 

ELECTRICAL  ENGINEERING— PHYSICS 

7 — General  and  Unclassified;  Batteries;  Central  Station  Practice; 
Distribution  and  Transmission;  Dynamo-Electro  Machinery; 
Electro-Chemistry  and  Metallurgy;  Measuring  Instruments 
and  Miscellaneous  Apparatus. 


8 — Astronomy.      Meteorology.      Explosives.      Marine    and 
Naval  Engineering.     Military.    Miscellaneous  Books. 

MATHEMATICS 

9 — General;    Algebra;   Analytic  and  Plane   Geometry;    Calculus; 
Trigonometry;  Vector  Analysis. 

MECHANICAL  ENGINEERING 

lOa  GeneraJ  and  Unclassified;  Foundry  Practice;  Shop  Practice. 
lOb  Gas  Power  and    Internal   Combustion  Engines;  Heating  and 

Ventilation;  Refrigeration. 
lOc  Machine  Design  and  Mechanism;  Power  Transmission;  Steam 

Power  and  Power  Plants;  Thermodynamics  and  Heat  Power. 
11 — Mechanics.  

12 — Medicine.  Pharmacy.  Medical  and  Pharmaceutical  Chem- 
istry. Sanitary  Science  and  Engineering.  Bacteriology  and 
Biology. 

MINING  ENGINEERING 

13 — General;  Assaying;  Excavation,  Earthwork,  Tunneling,  Etc.; 
Explosives;  Geology;  Metallurgy;  Mineralogy;  Prospecting; 
Ventilation. 


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